THE GENERAL STRUCTURE OF EDGE-CONNECTIVITYOF A VERTEX SUBSET IN A GRAPHAND ITS INCREMENTALMAINTENANCE.ODD CASEYe. Dinitz�;12 and A. Vainshteiny;3� Department of Computer Science, TechnionHaifa, Israel 32000, dinitz@cs.technion.ac.ily Dept. of Mathematics and Dept. of Comp. Science, Univ. of HaifaMount Carmel, Haifa 31905, Israel, alek@mathcs2.haifa.ac.ilOctober 26, 1997Abstract. Let G = (V;E) be an undirected graph, S be a subset of itsvertices, CS be the set of minimum edge-cuts partitioning S, �S be the car-dinality of such a cut. We suggest a graph structure, called the connectivitycarcass of S, that represents both cuts in CS and the partition of V by allthese cuts; its size is O(minfjEj; �SjV jg). In this paper we present gen-eral constructions and study in detail the case �S odd; the speci�cs of thecase �S even are considered elsewhere. For an adequate description of theconnectivity carcass we introduce a new type of graphs: locally orientablegraphs, which generalize digraphs. The connectivity carcass consists of alocally orientable quotient graph of G, a cactus tree (in case �S odd, justa tree) representing all distinct partitions of S by cuts in CS, and a map-ping connecting them. One can build it in O(jSj) max- ow computationsin G. For an arbitrary sequence of u edge insertions not changing �S, theconnectivity carcass can be maintained in time O(jV jminfjEj; �SjV jg + u).For two vertices of G, queries asking whether they are separated by a cut inCS are answered in O(1) worst-case time per query. Another possibility is tomaintain the carcass in O(jSjminfjEj; �SjV jg + u) time, but to answer thequeries in O(1) time only if at least one of the vertices belongs to S.1. IntroductionConnectivity problems play an important role in applications of graphtheory in computer science and have been extensively studied. In particu-lar, much attention has been given to edge connectivity problems. A pair ofvertices s and t of an undirected graph G = (V;E) is said to be k(-edge)-connected if there exist k edge-disjoint paths between them (equivalently,there is no k0-cut, k0 6 k � 1, separating s and t). The k-connectivity is1This research was supported in part by the Fund for the Promotion of Research at theTechnion, Israel.2Up to 1990: E.A.Dinic, Moscow.3The research of this author was supported by the Rashi Foundation.Typeset by AMS-TEXTec

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2an equivalence relation, and its equivalence classes are called k-connectivityclasses. It is known that the system of globally minimum cuts and connec-tivity classes formed by them in a graph is represented by a cactus tree,i.e., a graph whose blocks are edges and cycles [DKL]; it is just a tree ifthe size of globally minimum cuts is odd. On the other hand, the latticeof all (s; t)-minimum cuts for �xed vertices s and t is represented by allclosed sets of some directed acyclic graph (dag) with one source and onesink [PQ]4. Both these data structures can be maintained e�ciently underedge insertions (see [DW] for the �rst one and, e.g., [I] for the second). In[DV1] we have suggested a new structure, which is a natural generalization ofboth the aforementioned representations. This data structure represents allminimum cuts of G partitioning a �xed vertex subset and the correspondingpartition of this subset into its connectivity classes; besides, it represents thepartition of V by all these cuts. Our structure admits e�cient incrementalmaintenance as well.In this paper we describe general constructions and provide a detailedanalysis of the odd case, that is, of the case when the cardinality of thecuts in question is odd. This allows us to introduce all the main ideas whileavoiding many technical di�culties. A detailed analysis of the even case isgiven in [DV3].A well-known model for pairwise minimum cuts in a graph is the Gomory-Hu tree [GH]. It represents one minimum (x; y)-cut for any pair of verticesx; y, which is far from being su�cient for the purpose of incremental main-tenance. Indeed, if the newly inserted edge increases the value of the singleminimum (x; y)-cut represented by the Gomory-Hu tree, one cannot tellwhether the size of the minimum (x; y)-cut remains the same, or increases.One of the goals of connectivity studies is to provide tools for fast answer-ing connectivity queries in dynamic graphs, such as: \Are vertices u and vk-connected?", or \Show k-cuts separating u and v". An important prob-lem of this type is maintaining of the hierarchy of the k-connectivity classes,k 6 l; the corresponding incremental algorithms (that allow insertions ofedges to G) are presented in [WT] for l = 2, [GI] for l = 3, and [DW] forl = 4. We hope that our data structure can play the crucial role in solvingthis problem for arbitrary l. Indeed, take for a vertex subset a (k � 1)-connectivity class, then its connectivity subclasses are just k-connectivityclasses of G. The totality of these subclasses gives the partition of V intok-classes. Thus, it su�ces to maintain our data structures for all (k � 1)-classes of the graph, k 6 l. Observe that when an edge insertion causes themerging of some (k�1)-classes, their connectivity structures must be mergedas well. So, the results of this paper cover the \local" part of the work onthe maintenance of the hierarchy of connectivity classes for a general l.Another application of connectivity structures is augmentation problems4This fact was discovered independently by A. Karzanov (personal communication,1977).Tec

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3of the type: given a graph, add the minimum number of edges so as toincrease the connectivity of the whole graph to a prescribed value. Thecactus structure of minimum edge-cuts was employed essentially in [NGM,Be] to develop clear and fast schemes and algorithms for graph augmentation.One can suppose that the suggested generalization of this structure to locallyminimum cuts will help to improve the results in this area.Finally, the analysis of the connectivity structure of a vertex subset ina graph can arise itself in an application. For example, let us consider aninterconnection network N whose nodes are terminals and auxiliary inter-mediate stations. Then it is natural to de�ne the connectivity structure ofNas the connectivity structure of the set of terminals in N . As an illustration,consider the network with the terminals A, B and C shown on Fig. 1.1; the3-cuts shown by dashed lines are fA;B;Cg-mincuts, but there exist also twoirrelevant 2-cuts and one 3-cut.A C

BFig. 1.1. Minimum cuts in an interconnection networkLet us consider a vertex subset S and the set of all minimum S-cuts,that is, cuts of G that partition S and have the minimum cardinality, �S ,among such cuts; �S is said to be the connectivity of S. Two such cutsare equivalent if they partition S in the same way. The system of all suchpartitions, and thus all the (�S +1)-connectivity classes of S, is representedby a cactus tree HS (the skeleton), similarly to [DKL]. Partitions are de�nedby cuts of HS almost bijectively. In the odd case the skeleton is just a tree,and the representation of the partitions by its cuts is a bijection.Each family of equivalent S-cuts is represented, similarly to [PQ], by cutsof a two-terminal dag of a special type, called a strip, which is de�ned up tothe reversal of all its edges. Such a strip is a quotient graph of G; we identifyits vertices with their preimages in V under the quotient mapping. Thesestrips agree on intersections in the following sense. First, if two nonterminalvertices of two strips intersect (as subsets of V ), then these vertices coin-cide. Thus, the total partition of V by all minimum S-cuts subdivides onlyterminal vertices of each strip. The quotient graph by this total partitionwe denote by FS . Second, let two vertices of two strips coincide, and hence,these vertices have the same set of incident edges. Then the two 2-partitionsof this set into incoming and outcoming edges in both strips coincide as well.So, at each vertex there exists a unique distinguished 2-partition of the setTec

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4of incident edges (the inherent partition). The graph FS with the inherentpartitions at its vertices we call the esh.We introduce and study a special type of graphs: undirected graphs witha 2-partition of the set of incident edges at each vertex; we call them lo-cally orientable graphs. In general such a graph cannot be oriented globally,but still preserves certain properties of digraphs. An analog of an orientedpath in a locally orientable graph is a path that agrees with inherent parti-tions (a coherent path); we consider reachability along coherent paths. Forgraphs satisfying a natural restriction on cyclic coherent paths one can de-�ne strongly connected components and adjust Depth-First Search (DFS) toscanning such graphs and �nding these components. The esh FS satis�eseven a stronger restriction: all its coherent paths are simple; thus, FS is, ina sense, acyclic.Finally, to each vertex of FS we assign the set of families of equivalentS-cuts in whose strips this vertex is nonterminal. The corresponding cuts ofHS de�ne a subset of edges in HS (the projection �S of the vertex), whichturns out to be a path. The structure consisting of the skeleton, the esh,and the projection mapping is called the connectivity carcass of S.The connectivity carcass allows to answer readily several types of queries.The query asking whether two given vertices in S are (k + 1)-connected isanswered just by checking whether they are mapped into the same node ofHS or whether they are contained in the same vertex of FS . In the latterway one can check also whether two arbitrary vertices of G are separatedby an S-cut. An S-cut separating two vertices of which at least one belongsto S is de�ned immediately in terms of their projections. The same queryin the general case may require, in addition, to combine such a cut witha reachability cone of one of the given vertices. For any two subsets S1,S2 of S, the corresponding strip of minimum S-cuts separating S1 fromS2 is obtained by a contraction of the esh de�ned in a direct way by theprojection mapping. The orientations of the edges in this strip are inducedby the inherent partitions at the vertices of the esh.The connectivity carcass can be naturally \glued" from (s; t)-strips oftype [PQ], s; t 2 S, s �xed, by a polynomial algorithm, because of their\coincidences on intersections". The space requirements are linear, sincethis data structure is based on a quotient of the original graph.When considering incremental dynamics of the connectivity carcass, weassume that the cardinality of S-mincuts does not change under edge in-sertions; if such a change occurs, we say that the the connectivity carcassvanishes. As a matter of fact, there arises a new system of larger S-mincuts,but this transition is beyond the scope of dynamic considerations in thispaper.Upon inserting a new edge into G, all the three components of the car-cass change accordingly, and all the changes are of a contractive nature(provided �S does not change). The change of the graph of the esh con-Tec

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5sists in contracting several vertices into a single new vertex. In particular,the endpoints U1 and U2 of the image of the new edge in FS , and all thevertices that can be visited by a coherent path between U1 and U2, \fall"into the new vertex. The other esh vertices that are contracted are de�nedin terms of projections and reachability. The inherent partitions remain thesame, except for the new esh vertex, where the inherent partition is gluednaturally from those of constituting vertices, or becomes trivial in case itwas trivial for at least one of the constituting vertices. The change of theskeleton consists in contracting edges and \squeezing" cycles in the path-of-edges-and-cycles connecting the projections of U1 and U2; in the odd case itis just an ordinary path, and all of its edges are contracted. The projectionof the new esh vertex is �S(U1)\�S(U2). The set of the other esh verticesfor which the projection changes is de�ned by their reachability from U1 andU2. From the projection of a vertex, certain parts of the sets �S(U1)n�S(U2)or �S(U2) n �S(U1) are deleted.The incremental algorithm [DV1] for maintaining the connectivity car-cass is built according to the above scheme. In particular, it maintains thereachability cones of units, and thus involves, as a subproblem, incremen-tal maintenance of transitive closures. The complexity of this algorithm isO(jV jminfjEj; �SjV jg+ u), where u is the number of edge insertions (it isassumed that all of them preserve the connectivity of S); therefore, we donot exceed the complexity of best known algorithms for maintenance of thetransitive closure. Each query asking whether two given vertices are sepa-rated by a minimum S-cut is answered in O(1) worst-case time; such a cutitself can be shown in O(jV j) amortized time. The dag representation of allminimum S-cuts separating any two subsets of S can be obtained in O(jEj)amortized time. (For a comparison, in [GN] such a representation for anarbitrary pair of one-element subsets is constructed in O(jV j � jEj) time.)The complexity of maintenance can be reduced at the expense of an in-crease in the reaction time for certain queries: we guarantee the same O(1)worst-case time only in the cases when at least one of the vertices in ques-tion belongs to S. To avoid the time consuming maintenance of reachabilitycones, we suggest (see [DV2]) to maintain, instead of the actual esh, aweaker contraction of the initial esh, called the pre esh; the actual eshis obtained from the pre esh by contraction of its strongly connected com-ponents. The skeleton, the pre esh, and the projection mapping can bemaintained incrementally in time O(jSjminfjEj; �SjV jg + u) for an arbi-trary sequence of u edge insertions preserving the connectivity of S. Theonly increase of the reaction time, to O(jEj) worst case, arises for the sep-aration query in the case when the vertices in question both do not belongto S and have the same projection; the cut query in this situation can beanswered within the same O(jEj) time. As a corollary, we can simulta-neously maintain the carcasses for all parts of an arbitrary partition of Vin O(jV jminfjEj; �SjV jg + u�(u; jV j)) time. The same complexity can beTec

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6achieved, thus, for all the local work while maintaining the hierarchy ofpartitions of V into the k-connectivity classes, k 6 l.The �rst part of the paper treats the general case (�S arbitrary). InSect. 2 we give basic de�nitions and concepts. In Sect. 3 we introduce locallyorientable graphs and study their properties. In Sect. 4 we consider a numberof simple cases to gain intuition and to prove several basic lemmas. In Sect. 5we de�ne the esh (for the general case) and the skeleton (for the odd caseonly) and state their properties. In the rest of the paper �S is assumed tobe odd. In Sect. 6 we de�ne the projection and present further propertiesof the connectivity carcass. In Sect. 7 we describe the transformations ofthe connectivity carcass under the insertion of an edge to G. In Sect. 8we describe algorithms for construction and incremental maintenance of theconnectivity carcass and for answering the queries.Some results of this paper were published in a preliminary form in [DV1,DV2]. 2. Basic definitions and propertiesIn this paper we consider connected undirected graphs G = (V;E), jV j >2, without loops and possibly with parallel edges. We assume that theseproperties are preserved under vertex contractions; that is, we delete theobtained loops, but keep all the parallel edges. As usually, we denote jV j byn and jEj by m. For each ordered 2-partition P = (VP ; �VP), �VP = V n VP ,of the vertex set V we de�ne its edge set EP as the set of edges whose endslie in distinct parts of P . An ordered 2-partition C is said to be a cut of Gif no proper subset of EC coincides with EP for some other 2-partition P .The sets VC and �VC are called the sides of C; we say that a vertex v 2 V liesinside C if v 2 VC, and outside C if v 2 �VC. For any cut C, the opposite cut �Cis de�ned by V �C = �VC, �V �C = VC ; we say that �C is obtained from C by ipping.Let C and C0 be two cuts such that VC � VC0; we say that C0 dominates Cand write C � C 0. If the inclusion VC � VC0 is strict, we say that C 0 strictlydominates C and write C � C0.For any two cuts C and C 0, denote by C \C0 the 2-partition (VC \VC0 ; �VC [�VC0), and by C [ C0 the 2-partition (VC [ VC0; �VC \ �VC0). Observe that the2-partitions C \ C 0 and C [ C 0 are not necessarily cuts; however, if they are,then C \C0 � C � C [C 0 and C \C0 � C 0 � C [C0. As usually, the cardinalityc(C) = c(VC; �VC) of a cut C is de�ned to be the size of its edge set EC; clearly,c( �C) = c(C). The notion of cardinality is extended naturally to arbitrary 2-partitions of V , and moreover, to arbitrary pairs of disjoint subsets of V : ifX1; X2 � V , X1 \X2 = ?, then c(X1; X2) stands for the number of edgeswith one end in X1 and the other in X2.Let S be a subset of V ; we denote by � the cardinality of S. Any cut Cde�nes a 2-partition (SC; �SC) of S, with SC = VC \ S, �SC = �VC \ S. A cutC is said to be an S-cut if both SC and �SC are nonvoid. For S = fa; bg, wespeak of (a; b)-cuts instead of fa; bg-cuts, to keep the usual notation. It isTec

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7easy to see that the minimum cardinality among 2-partitions dividing S isequal to that among S-cuts; we denote it by �S . Moreover, each 2-partitionof minimum cardinality �S dividing S is an S-cut. In this paper we areinterested only in S-cuts of cardinality �S; we call them minimum S-cuts,or S-mincuts. The following basic property of S-mincuts (see Fig. 2.1) isused extensively throughout the paper._

_.

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CFig. 2.1. Two intersecting S-mincutsLemma 2.1. Let C and C0 be two arbitrary S-mincuts such that both VC \VC0 \ S and �VC \ �VC0 \ S are nonempty. Then(i) c(VC \ �VC0; �VC \ VC0) = 0;(ii) c(VC \ VC0 ; VC \ �VC0) = c(VC \ �VC0 ; �VC \ �VC0), c(VC \ VC0; �VC \ VC0) =c( �VC \ VC0 ; �VC \ �VC0).Proof. (i) We assume that C 6= C 0, since otherwise the assertion is trivial.Let s and t be two arbitrary vertices in VC \ VC0 \ S and �VC \ �VC0 \ S,respectively. Then both C and C 0 are (s; t)-mincuts, and hence, by [FF,Ch. 1, Corollary 5.3], every edge e between VC \ �VC0 and VC0 \ �VC must beboth free of any maximal (s; t)- ow and saturated by it.(ii) By (i), for any maximal (s; t)- ow c(VC \ VC0 ; VC \ �VC0) is the owentering VC \ �VC0, and c(VC \ �VC0 ; �VC \ �VC0) is the ow leaving VC \ �VC0X \ �Y .These two quantities are equal by the ow conservation law. The secondequality is proved in the same way. �The following observation, though trivial, is important: for any two S-mincuts C1 and C2, at least one of the pairs C1; C2 and C1; �C2 satis�es theconditions of Lemma 2.1.Two S-cuts are said to be S-equivalent if they de�ne the same 2-partitionof S. Equivalence classes of S-mincuts are called bunches . Evidently, if onereplaces each cut of a bunch by the opposite cut, then one again gets a bunch,which is said to be opposite to the initial one. The following statement is animmediate corollary of Lemma 2.1.Fact 2.2. Let C and C0 be two S-mincuts satisfying the conditions of Lemma2.1. Then:(i) Both C \ C0 and C [ C 0 are S-mincuts.(ii) Let C 00 be an S-mincut S-equivalent to C0, then C \ C00 and C [ C00 areS-mincuts S-equivalent to C \ C 0 and C [ C 0, respectively.Tec

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8In particular, the intersection and the union of two S-equivalent S-mincutsare cuts from the same bunch as well.Let S1 and S2 be two disjoint subsets of S. An S-mincut is said tobe an (S1; S2)-mincut if it keeps S1 and S2 on di�erent sides. It followsimmediately from Fact 2.2(i) that the intersection of all (S1; S2)-mincutshaving S1 inside is again an (S1; S2)-mincut. We call it the S1-tight (S1; S2)-mincut.Two vertices of G are said to be S-equivalent if they are not separated byany S-mincut. Equivalence classes of vertices are called S-units , or simplyunits . Evidently, the unit containing a given vertex is just the inner side ofthe intersection of all S-mincuts having this vertex inside. We denote by FSthe quotient graph obtained from G by contracting all the vertices of thesame unit. The vertices of FS are naturally called units. Each S-mincutinduces naturally a cut of FS separating the same units; in what follows wedo not distinguish between these two cuts.Observe that, in general, S is partitioned by the set of all S-mincuts intosubsets S1, S2; : : : ; Sr, and r can be less than �. In this case FS coincideswith Ffv1;v2;:::;vrg, where vi is an arbitrary vertex in Si, 1 6 i 6 r.Let f : V !M be a mapping to an arbitrary set M . For any W � V , wedenote by f(W ) the set ff(w) : w 2 Wg. We say that a 2-partition P of Mf -induces a cut C if VC = f�1(MP). Similarly, given a mapping g : S !M ,we say that P g-induces a bunch of S-cuts if SC = g�1(MP) for any cut Cin this bunch.To illustrate the above notions, let us consider the graph G presented onFig. 2.2a. Large black circles denote vertices in S; thus �S = 2. All the tenS-mincuts are shown by dashed lines. They fall into four bunches consistingof 1, 2, 2, and 5 cuts, respectively, marked by braces. The vertices of G fallinto eight units marked by numbers. The quotient graph FS is presentedon Fig. 2.2b. The quotient mapping f takes all the vertices of a unit ito the vertex i of FS . Each S-mincut is f -induced by a unique cut of FS(for example, see cuts C1 and C2 in Figs. 2.2a,b). Let us consider also afour-element set M = fa; b; c; dg and the mapping g : S ! M that takesthe vertices of S belonging to the units 2, 3, 4, 8 to a; c; b; d, respectively.The four 2-partitions of M shown on Fig. 2.2c g-induce the four bunches ofS-mincuts in G.Tec

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cdFig. 2.2. S-mincuts, the graph FS, and inducing of cuts andbunches 3. Locally orientable graphs3.1. General locally orientable graphs. Let v be an arbitrary vertexof an undirected graph. The star of v is the set of edges incident to it. Thestar of a vertex is said to be orientable if some its unordered 2-partition is�xed; the parts of this 2-partition are called the sides of the star. Such avertex is called stretched if the partition of its star is nontrivial (i.e., bothsides are nonempty) and a terminal otherwise. A graph is said to be locallyorientable if the stars of all its vertices are orientable; we say that it overliesthe initial undirected graph. A locally orientable graph is called balanced iffor any stretched vertex the two sides of its star have equal cardinalities.In particular, for any vertex v of an arbitrary directed graph, the partitionof arcs incident to v into incoming and outcoming de�nes a 2-partition of thestar of v in the underlying undirected graph. Thus, for any digraph, thereexists a canonically de�ned locally orientable graph, which underlies thisdigraph and overlies its underlying undirected graph. A locally orientablegraph is called globally orientable if it underlies a directed graph. Figure 3.1provides two examples of locally orientable graphs of which one is globallyorientable, while the other is not. Here and in what follows stretched ver-tices are denoted by rectangles to visualize the corresponding 2-partitions;terminals are denoted by circles. Globally orientable graphs are essentiallyequivalent to directed graphs in the following sense.

A globally orient-able graph

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10Lemma 3.1. Given a connected globally orientable graph, the overlying di-rected graph is restored uniquely, up to the global reversal of arcs.Proof. Let ~G1 and ~G2 be two digraphs overlying a given locally orientablegraph G. Let V + be the subset of vertices at which the ordered 2-partitionsof the star into incoming and outcoming arcs in ~G1 and ~G2 coincide, and V �be the subset of vertices at which they are opposite. Evidently, (V +; V �)is a partition of V . Assume that both its parts are nonempty. Since G isconnected, it contains an edge between vertices v+ 2 V + and v� 2 V �.Assume, w.l.o.g., that the corresponding arc in ~G1 leaves v+ and enters v�.Then the corresponding arc in ~G2 must leave both of them, a contradiction.Thus either V � = ? and ~G1 �= ~G2, or V + = ? and ~G2 is opposite to ~G1. �The following construction provides a criterion for the global orientabilityof a locally orientable graph G (we do not use this criterion in what follows).Let us split each vertex v of G and its star into the two vertices v1 and v2whose stars are exactly the sides at v and add the auxiliary edge (v1; v2);the obtained graph we denote G0.Proposition 3.2. A locally orientable graph G is globally orientable if andonly if G0 is bipartite.Proof. Let G be globally orientable and ~G be its overlying digraph. Letus transform ~G into ~G0 similarly to the above transformation; evidently, ~G0overlies G0. The partition (fv1 : v 2 V g; fv2 : v 2 V g) shows that G0 isbipartite.Let now G0 be bipartite with the partition (V �; V ��). Let us direct alledges from V � to V ��. The contraction of all auxiliary edges results in adigraph overlying G. It is consistent with the 2-partitions at vertices sincefor each v 2 V the corresponding vertices v1 and v2 belong to di�erent parts,i.e., arcs only leave v1 and only enter v2, or vice versa. �According to Lemma 3.1, locally orientable graphs are a generalization ofdirected graphs. Some concepts of digraphs have their natural analogs forthis generalization. A path (v0; e1; v1; e2; : : : ; er; vr) in a locally orientablegraph is said to be coherent if for any of its inner vertices vi, 1 6 i 6 r � 1,the edges ei and ei+1 belong to the opposite sides of the star of vi. Clearly,the inverses of any coherent path, any single edge path, and any single vertexpath are coherent paths. Observe that each coherent path can be extendedbeyond its endpoint, provided it is not a terminal. A cyclic coherent path(for which v0 = vr and r > 1) is called a coherent cycle if the edges er and e1belong to the opposite sides at v0 (see Fig. 3.2). Coherent paths and cyclesare natural analogs of directed paths and cycles in digraphs. In particular, acoherent path in a globally orientable graph corresponds to a directed pathin the overlying digraph or to the inverse of such a path.Tec

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11A coherent path A coherent cycle Not a coherent cycle

. Fig. 3.2. Coherentness of paths and cyclesWe call a 2-partition P of the vertex set of a locally orientable graph(in particular, its cut) transversal if any coherent path intersects the cor-responding edge set EP at most by one edge. Thus, if both endpoints ofa coherent path lie on the same side of a transversal cut, then all the pathlies on the same side of this cut. Observe that the edge set of a transversal2-partition intersects at most one side of the star of any vertex. Moreover,the following statement can be proved easily.Fact 3.3. Let P1 and P2 be two transversal 2-partitions of the vertex set ofa locally orientable graph such that VP1 \ VP2 is a single stretched vertex v.Then the intersections of the star of v with the edge sets of P1 and P2 areexactly the two sides of this star.A locally orientable graph is called coherent if each its cyclic coherentpath is a coherent cycle. For a coherent locally orientable graph, all verticesand edges of coherent paths beginning at a vertex v are called reachablefrom v. All these vertices and edges form the reachability subgraph of v.Observe that the reachability relation on vertices is symmetric (though nottransitive: see vertices x, y, z in Fig. 3.1a). Therefore, the notion of mutualreachability of two vertices is well de�ned.Lemma 3.4. The reachability subgraph of any vertex of a coherent locallyorientable graph is globally orientable.Proof. To construct a global orientation of the reachability subgraph of vwe proceed as follows. Let us choose a side of the star of v. Moving alongany coherent path starting at v and leaving v from this side, we direct all itsedges from v. Similarly, moving along any coherent path starting at v andleaving v from the opposite side, we direct all its edges to v.Let us prove the following statement: if two paths P1 and P2 from v toan arbitrary vertex u leave v from the same side (from the opposite sides),then they enter u at the same side (at the opposite sides). Assume to thecontrary that P1 and P2 leave v from the same side and enter u at theopposite sides (see Fig. 3.3a). Then the cyclic coherent path composed ofP1 and the inverse of P2 is not a coherent cycle. Similarly, if P1 and P2leave v from the opposite sides and enter u at the same side, then the cycliccoherent path composed of the inverse of P2 and P1 is not a coherent cycle(see Fig. 3.3b).Tec

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uvvFig. 3.3. Cases in the proof of Lemma 3.4Assume now that in our construction some edge e acquires two oppositeorientations via paths P1 and P2. Evidently, the behavior of these paths atan arbitrary endpoint of e distinct from v contradicts to the above statement.Thus, the orientation of edges is well de�ned. Finally, to prove its consistencywith the initial orientability structure we have to ensure that if two edgese1; e2 incident to a vertex u are oriented to and from u, respectively, thenthey are at the opposite sides of u. For u = v it is evident. For u 6= v, wehave to consider four cases depending on the types of the paths P1 and P2that de�ned the orientations of the edges e1; e2, respectively. For example,let both P1 and P2 leave v from the side opposite to the one chosen in theconstruction and let e1 and e2 be oriented as in Fig. 3.3c. Then P2 entersu via e2 while P1 enters u via an edge preceding e1 in P1. Thus P2 and P1enter u at the opposite sides, a contradiction with the above statement. Thethree other cases are handled in the same way. �According to Lemmas 3.1, 3.4, one can construct in an obvious way aversion of the Depth-First Search (DFS) that starts from an arbitrary vertexv and scans all the vertices and edges reachable from v in the same direction(i.e., by paths leaving v from the same side).For a coherent locally orientable graph, the relation \two vertices belongto the same coherent cycle" is symmetric and transitive; the subgraphs in-duced by its equivalence classes are naturally called strongly connected com-ponents. It is easy to see that strongly connected components intersecting areachability subgraph lie entirely in this subgraph and correspond bijectivelyto the strongly connected components of its overlying digraph. Hence, wecan execute any DFS-based linear algorithm for �nding strongly connectedcomponents on any reachability subgraph.An analog of a dag is an acyclic locally orientable graph, the one withoutcyclic coherent paths; thus, it is coherent. In particular, the underlyinggraph of a dag is acyclic in the above sense. As in the proof of Lemma 3.4,it follows from the acyclicity that for any two coherent paths joining v1and v2, their initial edges leave v1 from the same side. Thus, the set ofvertices reachable from v1 splits into two reachability cones correspondingto the sides of its star (if v1 is a terminal, one of the cones is trivial). Itis easy to see that if v2 belongs to a reachability cone R of v1, then one ofthe reachability cones of v2 contains v1 while the other lies strictly inside R(this property may be regarded as a weak analog of transitivity). Acyclicityimplies that each coherent path can be extended beyond each of its endpointsT

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13up to a terminal. As a corollary, each nontrivial reachability cone of a vertexcontains at least one terminal distinct from this vertex.3.2. Strips. In this subsection we consider acyclic two-terminal locallyorientable graphs.Lemma 3.5. Any acyclic two-terminal locally orientable graph is globallyorientable.Proof. By Lemma 3.4, it is enough to prove that the reachability subgraphof a terminal in such a graph coincides with the entire graph. Each vertexor edge is a coherent path; therefore, it can be extended to a coherent pathbetween two terminals. By the acyclicity, these terminals are distinct, andthe result follows. �According to this Lemma, properties of an acyclic two-terminal locallyorientable graph G are similar to those of its overlying dag ~G. In particular,the following statements are true.Fact 3.6.(i) The transversal cuts of G underlie the cuts of ~G in which all arcs aredirected from the part containing the source to the part containing the sink.(ii) A transversal 2-partition of G is a (transversal) cut separating theterminals.(iii) A reachability cone of a vertex in G de�nes a transversal cut ; it is theintersection of all transversal cuts such that both this vertex and the terminalthat belongs to this cone lie inside them.An acyclic two-terminal globally orientable graph is called a strip (ofwidth w) if all its transversal cuts are of the same cardinality w. Observethat the degree of a terminal in a strip equals its width (since the 2-partitionisolating this terminal is a transversal cut).Let s and t be the terminals and v be a vertex of a strip. We label thesides of the 2-partition and the cones at v by s and t; the cones are denotedRs(v) and Rt(v). Thus, any two coherent paths (u1; : : : ; u2) 2 Rx(u1) and(u2; : : : ; u3) 2 Rx(u2), x = s or t, can be concatenated into a coherent path(u1; : : : ; u2; : : : ; u3) 2 Rx(u1).Strip Lemma.(i) An acyclic two-terminal locally orientable graph is a strip if and onlyif it is balanced.(ii) There exists at most one strip for a �xed underlying graph and a �xedpair of terminals.Proof. (i) Let G be a balanced acyclic locally orientable graph with twoterminals s and t. By Lemma 3.5, G is globally orientable. Let ~G be theoverlying digraph of G in which s is the source and C be a transversal cutof G such that s 2 VC. We consider the di�erence � between the sum ofTec

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14outdegrees (in ~G) for all vertices in VC and the sum of their indegrees. On onehand, since G is balanced and t =2 VC, � equals the outdegree of s. On theother hand, since each internal arc of VC contributes 1 to both sums and allexternal arcs are directed outside (by Fact 3.6(i)), � equals the cardinalityof EC. Thus c(C) equals the outdegree of s for any transversal cut C.Let now G be a strip and ~G be its overlying dag with the source s and thesink t. Let v be an arbitrary nonterminal vertex and R be the reachabilitycone of v in ~G. Both 2-partitions (R; V n R) and (R n v; (V n R) [ v) aretransversal since any directed path can only enter R (or Rnv) but not leaveit. By Fact 3.6(ii), they are cuts. Their cardinalities di�er by the di�erencebetween the indegree and the outdegree of v. Since both cardinalities areequal, the indegree and outdegree at v are equal as well.(ii) The statement is evident for graphs with two vertices; hence, weassume that there exists a vertex v 6= s; t. Let us consider a strip G with asource s and the sink t. It is easy to see that the deletion of s from G yieldsnew terminals. Indeed, the one-vertex coherent path v can be extended upto terminals in both directions; these terminals are distinct since the graphis acyclic. As it was shown in the proof of Lemma 3.4, all the deleted edgesincident to an arbitrary vertex are at the same its side. Hence, for any newterminal exactly a half of incident edges is deleted, and for any other vertexstrictly less than a half (a characterization in terms of the underlying graph).Let us prove that the contraction of the new terminals implies a strip. Theobtained graph is evidently balanced and two-terminal. Assume that thereexists a cyclic coherent path in it. Evidently, it starts and ends at the newterminal and thus can be extended to a cyclic coherent path starting andending at s in the initial strip, a contradiction. Finally, the initial strip canbe restored uniquely from the strip obtained.Assume to the contrary that assertion (ii) is not valid. Then there existsa minimal counterexample: a graph with the minimal number of verticesgiving rise to two distinct strips. Let us execute with these strips the aboveprocedure. According to the characterization mentioned, the resulting undi-rected graphs coincide. Thus, by the minimality of the counterexample, theresulting strips coincide as well. By the remark at the end of the previousparagraph, the same holds for the initial strips, a contradiction. �The concept of a strip allows us to reformulate and re�ne the main resultof [PQ] on the representation of all minimum cuts between two vertices ofa graph. Let us consider an undirected graph G and an arbitrary pair ofits vertices a; b. Assume f be a maximal ow from a to b in G, and Gfbe the corresponding residual graph. The result of the contraction of eachstrongly connected component of Gf to a new supervertex is a dag Da;b notdepending on f . The corresponding quotient mapping is denoted by �a;b,and the underlying locally orientable graph of this dag byWa;b (see Fig. 3.4).Tec

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15G Gf D a,ba,b WFig. 3.4. Construction of the (a; b)-stripTheorem 3.7. Let G be an undirected graph and a; b be a pair of its ver-tices. Then the graph Wa;b overlies Ffa;bg, is a strip, and its transversal cutscorrespond bijectively to the minimum (a; b)-cuts of G via the �a;b-inducing.Proof. By [PQ] the �a;b-inducing provides a bijection between the minimum(a; b)-cuts of G and the 2-partitions of type fa closed set of Da;b, its com-plementg. Recall that a vertex subset X in an arbitrary dag is said to beclosed if no other vertices can be reached from this subset, or, equivalently,if no arc goes from X to �X. According to Fact 3.6(i), the 2-partitions of theabove type overlie exactly the transversal cuts of Wa;b. Since all minimum(a; b)-cuts of G have the same cardinality, so do all the transversal cuts ofWa;b, and hence Wa;b is a strip. It follows from the above discussion that thepartition of V into the vertices of Wa;b is a re�nement of the partition intothe vertices of Ffa;bg. To prove that they, in fact, coincide, it is su�cientto show that each vertex U of Wa;b is separated from all other vertices bytransversal cuts. Indeed, by Fact 3.6(iii), these cuts are just the two cutsde�ned by reachability cones of U . �Remark. Observe that natural analogs of the above construction and of The-orem 3.7 are valid for directed graphs.The strip Wa;b will be referred to as the (a; b)-strip, and its verticesas (a; b)-units, or units of Wa;b. Obviously, Wb;a coincides with Wa;b and�b;a = �a;b. The above constructions and results can be extended easily tothe case of minimum cuts separating two disjoint vertex subsets A and B:it is enough to contract A and B into new supervertices. The corresponding(A;B)-strip is denoted by WA;B and the quotient mapping by �A;B .Lemma 3.8. Let u be a vertex of G and U 6= �a;b(b) be the corresponding(a; b)-unit. The contraction of the reachability cone of U in Wa;b containingthe terminal �a;b(a) gives the (fa; ug; b)-strip; its width is equal to the widthof the initial strip Wa;b.Proof (see Fig. 3.5). It is easy to see that the contraction of a side of atransversal cut in an acyclic locally orientable graph preserves acyclicity.Therefore, by Fact 3.6(iii) and Strip Lemma, the above contraction yieldsa strip W . Since the terminal �a;b(b) remains untouched, the width of Wequals that of Wa;b.T

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16a b

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C CuU’Fig. 3.5. To the proof of Lemma 3.8By Strip Lemma(ii), it is enough to prove now that the underlying graphof W coincides with that of Wfa;ug;b. Since all (fa; ug; b)-cuts are (a; b)-cuts, we can study them (more exactly, their images) in Wa;b. First, byFact 3.6(iii), the reachability coneR in question coincides with the (fa; ug; b)-unit containing fa; bg. Second, let U 0 and U 00 be two (a; b)-units not belong-ing to R and let C be an (a; b)-cut separating them. Evidently, the union ofC and the cut de�ned by R is an (fa; ug; b)-cut Cu separating U 0 and U 00. �Observe that for an arbitrary vertex subset S, the set of all minimum S-cuts is the union of the sets of minimum (a; b)-cuts for certain pairs of verticesa; b 2 S. It is natural to ask whether it is possible to glue the corresponding(a; b)-strips together to form a single object. The main �nding of this paperis that such an object exists: it is FS with a canonical structure of localorientability de�ned by these (a; b)-strips.4. Simple casesTo make our ideas more intuitive, let us consider separately several simplecases.4.1. A two-element subset S. From the statical point of view, the case� = 2 is covered completely by Theorem 3.7; recall that for S = fa; bg onehas FS =Wa;b.Let us turn to dynamics. In what follows, we put hat over any notation(e.g., bRc, or bFS) to denote the corresponding object after edge insertion. Re-call that the units of FS are de�ned as the strongly connected components ofthe graph Gf , where f is a maximum ow from a to b in G. Since we restrictourselves to the case of a �xed value of (a; b)-mincuts, we may use the same ow f for the entire dynamic process. Hence, we can maintain the graph Gf�xing local changes. Namely, the insertion of a new edge (u1; u2) implies theaddition of two arcs (u1; u2) and (u2; u1) to Gf . Moreover, one can maintainthe dag Da;b of the strongly connected components of Gf as follows. LetU1; U2 be the components containing u1; u2, respectively. The new dag bDa;bis obtained from the current one by adding arcs (U1; U2) and (U2; U1) andshrinking the new strongly connected component containing both U1 andU2. This transformation can be easily extended to the underlying stripWa;bas follows.T

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17If U1 = U2, then there are no changes in Wa;b. Otherwise, U1 and U2 arecontracted to a new unit Unew.If U1 and U2 are not mutually reachable in Wa;b, then there are no otherchanges inWa;b (see Fig. 4.1a). Otherwise, assume w.l.o.g. that U2 2 Rb(U1)and U1 2 Ra(U2); then Rb(U1) \ Ra(U2) (that is, the set of all the unitsand edges of all the paths between U1 and U2) is contracted into Unew aswell (see Fig. 4.1b).1

U2

1U

Ua b

b)

U2

a b2 1(U )(U )R R

a b

a)U new

U newFig. 4.1. Dynamics of Ffa;bgIf both U1 and U2 are terminals, then the system of S-mincuts vanishes.If exactly one of them is a terminal, then Unew is the corresponding terminalin the new strip. If both U1 and U2 are stretched, then Unew is stretched aswell, and the side of the 2-partition at Unew labeled by a (resp., b) is gluedfrom the sides of the 2-partitions at all the contracted units labeled by thesame letter (except for the internal edges of the contracted subgraph).Observe that the dag of the strongly connected components is exactly theobject maintained by algorithms [I,LL], with the time complexity O(mn).Evidently, any such algorithm can be easily modi�ed to maintain a strip,with the same complexity.The above description covers also a more general situation, when � isarbitrary, but all the S-mincuts divide S in the same way into two subsetsS1 and S2. We then consider these subsets as supervertices a and b and arriveat the above situation described completely by the strip WS1;S2 . Observethat in this case FS =WS1;S2 =Ws1;s2 = Ffs1;s2g for any s1 2 S1, s2 2 S2.4.2. A three-element subset S; the asymmetric subcase. The small-est nontrivial case, from the statical point of view, is S = fa; b; cg and anytwo of these vertices are separated by an S-mincut. Clearly, from the threepossible types of S-mincuts: (a; fb; cg), (b; fa; cg), and (c; fa; bg), at leasttwo must exist. So we have two subcases: the asymmetric, when exactlytwo types of S-mincuts are present, and the symmetric, when there arepresent all the three of them.Let us assume that there are no (b; fa; cg)-cuts. Thus, every S-mincutis an (a; c)-cut, and vice versa. Therefore, the entire family of S-mincutsis represented by the strip Wa;c; in particular, (a; c)-units coincide with S-units. We de�ne the structure of local orientability on the graph Ffa;b;cgby keeping that of Wa;c at all the units, except for the unit Ub containingb, which is now regarded as a terminal. Since Wa;c is acyclic and balanced,Tec

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18and turning a vertex to a terminal does not extend the set of coherent paths,we see that Ffa;b;cg is acyclic and balanced as well. Besides, we see that allS-cuts of G are transversal cuts in Ffa;b;cg; observe that the converse is nottrue, see, e.g., the cut that separates Ub from all the other units.We denote by (Lb; �Lb) the fa; bg-tight (fa; bg; c)-mincut. Similarly, (Rb; �Rb)is the fb; cg-tight (fb; cg; a)-mincut (see Fig. 4.2a). Observe that accordingto Lemma 2.1(i), there are no edges between Lb \ Rb and �Lb \ �Rb.__

LLb Rb

bca

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acW

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Fig. 4.2. Structure of Ffa;b;cg in the asymmetric caseIn our next statement we summarize other properties of the cuts (Lb; �Lb)and (Rb; �Rb).Lemma 4.1.(i) Lb and Rb are the two reachability cones of the unit Ub in Wa;c, andhence Lb \Rb coincides with this unit.(ii) The sides of the star of Ub in Wa;c are induced in Ffa;b;cg by edge-setsof (Lb; �Lb) and (Rb; �Rb).(iii) The contraction of Lb in Ffa;b;cg gives Wb;c, the contraction of Rb inFfa;b;cg gives Wa;b.Proof. (i) Follows from Fact 3.6(iii).(ii) Follows from (i) and Fact 3.3.(iii) Follows from Lemma 3.8. �Evidently, either Ub is a separating vertex of Ffa;b;cg, or Ffa;b;cg n Ubis connected. In the latter case we build from Ffa;b;cg a locally orientablegraph Wac by deleting all the vertices of Ub and contracting Lb \ �Rb to theterminal a0 and �Lb \Rb to the terminal c0 (see Fig. 4.2a). (Informally, Wacis the common part of Wa;b and Wb;c in Ffa;b;cg.) An S-mincut C of G thatseparates Lb \ �Rb from �Lb \ Rb and thus generates an (a0; c0)-mincut C0 inWac is called an extension of C 0.Lemma 4.2.(i) The graph Wac is a strip of width �0 = �S � 12 degUb < 12�S.(ii) All its transversal cuts, and only they, are extendable to S-mincuts.Tec

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19(iii) If a 2-partition of Lb [ Rb enters an extension of some transversalcut of Wac, then it enters some extension of any such cut. (Informally : alltransversal cuts of Wac are interchangeable in S-mincuts).Proof. (i) Observe that deletion of vertices does not give rise to new coherentpaths, thus, it preserves the acyclicity of a locally orientable graph. For thesame reason, any transversal 2-partition induces a transversal 2-partition inthe new graph. Since the contraction of a side of a transversal 2-partition pre-serves acyclicity, Wac is an acyclic two-terminal balanced locally orientablegraph. Hence, it is a strip by Strip Lemma(i). It is easy to see, applyingLemma 2.1, that �S = c(Lb; �Lb) = deg a0 + 12 degUb. Since �0 = deg a0,we get �0 = �S � 12 degUb. Assume that �0 > 12�S . Then degUb 6 �S ,which means that (Lb \ Rb; �Lb [ �Rb) is a 2-partition of cardinality at most�S separating b from fa; cg, a contradiction.(ii) Let us denote by W 0 the graph obtained from Ffa;b;cg by deleting Ub.By (i), any transversal cut (Z; �Z) of Wac has cardinality �0 and generates a2-partition (X; �X) of the same cardinality in W 0. By Lemma 2.1, the edgeset of the 2-partition (X [ Ub; �X) in Ffa;b;cg has exactly by 12 degUb moreedges, i.e., �0 + 12 degUb = �S edges. Thus, it is a cut, that is a sought forextension of (Z; �Z).Conversely, let (X; �X) be an S-mincut separating Lb \ �Rb from �Lb \ Rb.It generates a 2-partition of cardinality �0 in W 0 and a 2-partition of thesame cardinality separating a0 and c0 in Wac. Thus, it is a �0-cut separatinga0 and c0, i.e., a transversal cut.(iii) Since there are exactly two such partitions, namely, ((Lb \ �Rb) [Ub; �Lb \Rb) and (Lb\ �Rb; (�Lb\Rb)[Ub), one can proceed exactly as in theproof of (ii). �Remark. Observe that Lemma 4.2 remains true for a more general de�nitionof an extension, in which C is required only to separate the ends of all edgesbetween Lb\ �Rb and �Lb\ �Rb that lie in Lb\ �Rb and ends of all edges between�Lb \ �Rb and �Lb \ Rb that lie in �Lb \ Rb. Such a de�nition would allow, forexample, to consider the cuts C3 and C4 on Fig. 4.2b as extensions of C 0,while according to the current de�nition, the only extensions of C0 are C1and C2. However, in what follows we do not consider such a generalizationof the notion of extension.Dynamics of Ffa;b;cg in the asymmetric subcase are induced by those ofFfa;cg =Wa;c (see Sect. 4.1) straightforwardly, except for the case when Ub,and possibly some other units, is contracted to Unew. Let us consider thiscase; it occurs when Ub 2 Rc(U1) \ Ra(U2), up to ipping of U1 and U2(see Fig. 4.4). The only speci�cs in this case is that Unew becomes a newterminal (in fact, bUb).Tec

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a) b)

a= bc cU2U1

U2U1b

U UnewnewFig. 4.3. Dynamics of Ffa;b;cg in the asymmetric caseRemark. To translate the de�nition of the contracted set into the terms ofFfa;b;cg, one has to take into account that a coherent path in Wa;c is eithera coherent path in Ffa;b;cg, or a concatenation of two coherent paths: a pathto Ub in Lb and a path from Ub in Rb. Under this translation, the abovecondition on Ub is equivalent to U1 2 Ra(Ub) = Lb and U2 2 Rc(Ub) = Rb.There is a special situation when not only Ub, but Ua as well is con-tracted to Unew; this occurs when U1 = Ua (see Fig. 4.3b). In this situa-tion the asymmetric three-element case degenerates, and we arrive at thecase considered at the end of Sect. 4.1; that is, Ffa;b;cg becomes equal toWfa;bg;c = Fb;c = Fa;c. Evidently, Ua above can be replaced by Uc.Finally, if fU1; U2g = fUa; Ubg, then the system of S-mincuts vanishes.As we see, the reduction of the three-element case to the two-element caseis de�ned via subsets Lb and Rb. Let us consider the dynamics of Lb (thoseof Rb are similar) in the case when Ffa;b;cg does not degenerate.Lemma 4.3. The set Lb is extended if and only if U1 2 Lb and U2 =2 Lb(up to ipping of U1 and U2). The new part of bLb is the union of units U 0such that U 0 2 Ra(U2) and U 0 =2 Lb.Proof. The if part of the �rst statement is trivial, since U2 is added to Lb.Let U 0 be a new unit in bLb and let (U 0; : : : ; Unew; : : : ; Ub) be a coherentpath in cWa;c; clearly, it lies entirely in bRc(U 0). Its edges form two coherentpaths in Ffa;b;cg: (U 0; : : : ; U�) 2 Rc(U 0) and (U��; : : : ; Ub) 2 Rc(U��) (seeFig. 4.4). Since U 0 is new in bLb, we have U� 6= U��, and thus the state-ment holds for the case when Unew contains only U1 and U2. Otherwisewe may assume w.l.o.g. that U�; U�� � Unew = Rc(U1) \ Ra(U2). Hencein Wa;c exist coherent paths (U�; : : : ; U2) 2 Rc(U�) and (U1; : : : ; U��) 2Rc(U1). The �rst of them does not contain Ub by assumption, and thus(U 0; : : : ; U�; : : : ; U2) is a coherent path in Ffa;b;cg. Therefore, U2 2 Lb wouldimply U 0 2 Lb in a contradiction to the assumption that U 0 is new in bLb. Thepath (U1; : : : ; U��; : : : ; Ub) does not contain Ub twice (as a coherent path inWa;c), and thus is a coherent path in Ffa;b;cg; hence, U1 2 Lb. The assertionfollows. �T

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21Lb

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b

a 1U U2U

UU’

*

**Fig. 4.4. To the proof of Lemma 4.34.3. A three-element subset S; the symmetric subcase. Second, letall types of fa; b; cg-cuts exist in G. Let Cx = (Vx; �Vx) be the fy; zg-tight(fy; zg; x)-mincut (Fig. 4.5a); here and in what follows x; y; z 2 fa; b; cg,x 6= y 6= z. As usual, these three 2-partitions de�ne the following eightsubsets of V : V ? = Va \ Vb \ Vc; V a = �Va \ Vb \ Vc; V b = Va \ �Vb \ Vc;V c = Va\Vb\ �Vc; V ab = �Va\ �Vb\Vc; V bc = Va\ �Vb\ �Vc; V ac = �Va\Vb\ �Vc;V abc = �Va \ �Vb \ �Vc. These subsets are called cells.V

VV Va

bV

abV

V

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Vbc

b

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a,abC

C abc

caac

a

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b)

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a)

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ac cFig. 4.5. Structure of Ffa;b;cg in the symmetric caseThe following statement is crucial for the rest of our results (see Fig. 4.5b).3-Star Lemma.(i) V abc = ?.(ii) Vertices of V xy can be adjacent only to vertices of V xy ; V x; V y , andc(V xy ; V x) = c(V xy; V y).Proof. Let us consider the cut Cx;xy = Cz \ �Cx = (V x[V xy ; V n (V x[V xy)).Since both Cz ; Cx are minimum (x; z)-cuts, the cut Cx;xy is a minimum(x; z)-cut as well, and hence c(Cx;xy) = �S. Similarly, Cy;xy = Cz \ �Cy =(V y [ V xy ; V n (V y [ V xy)) is a minimum (y; z)-cut with c(Cy;xy) = �S .Therefore, both Cx;xy and Cy;xy are minimum (x; y)-cuts. According toLemma 2.1(i), there are no edges between V xy and V n (V x [ V xy [ V y),and moreover, c(V xy ; V x) = c(V xy; V y), so assertion (ii) is proved.We know already that there are no edges between V abc and V xy. ByLemma 2.1(i) applied to the minimum (x; y)-cuts Cx and Cy , we get thatthere are no edges between V xy [ V abc and V z [ V ?. So, there are noexternal edges incident to V abc at all. Since the initial graph is connected,we thus get V abc = ?. �Tec

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22Properties of the cuts Cx are similar to those of the cuts (Lb; �Lb) and(Rb; �Rb) (see Sect. 4.1). In particular, the �rst statement below can beregarded as an analog of Lemma 4.1(iii).Lemma 4.4.(i) Contraction of the inner side Vx of Cx in Ffa;b;cg gives the underlyingundirected graph of the strip Wfy;zg;x.(ii) Every unit, except for V ?, corresponds in this way to a vertex ineither one or two strips.(iii) For every unit U 6= V ?, the 2-partitions of its star induced by the2-partitions at the corresponding vertices in these strips coincide.Proof. (i) Since all (fy; zg; x)-mincuts are S-mincuts and Cx is the fy; zg-tight (fy; zg; x)-mincut, the result of the contraction still represents all(fy; zg; x)-mincuts. Therefore, the only thing we have to prove is that no(fy; zg; x)-unit distinct from the contracted one can be divided by an S-mincut. Indeed, let U , to the contrary, be such a unit and C be an S-mincutdividing U . Without loss of generality we can assume that C is an (z; fx; yg)-cut containing z inside; recall that z lies also inside Cx (see Fig. 4.6). ThenCx [ C is a (fy; zg; x)-mincut dividing U , a contradiction.C

U

y

x z

CUCC xxFig. 4.6. To the proof of Lemma 4.4(i)(ii) Follows from (i) immediately.(iii) Assume without loss of generality that U lies in V xy . According to(i) and (ii), U is a unit in the strips Wfy;zg;x and Wfx;zg;y. By Lemma 3.8,both these strips are contractions of the same strip Wx;y. Moreover, in bothcases U is not a contracted unit, so local orientations at U in both cases areinherited from Wx;y, and thus coincide. �Let us assign to every unit, except for V ?, the 2-partition of its starde�ned in Lemma 4.4(iii), and to V ? the trivial 2-partition. Then Ffa;b;cgbecomes a locally orientable graph. Evidently, the contraction of Ffa;b;cgdescribed in Lemma 4.4(i) yields now the strip Wfy;zg;x itself. One can say,thus, that all the strips Wfy;zg;x \are glued" into the aggregate structureFfa;b;cg because they \coincide on intersections".Lemma 4.5.(i) The locally orientable graph Ffa;b;cg is balanced and acyclic.(ii) Each S-mincut is transversal in Ffa;b;cg.T

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23Proof. (i) Evidently, Ffa;b;cg is balanced, since the 2-partition at each non-terminal unit is inherited from a strip. Let us prove its acyclicity. Indeed, letthere exist a cyclic coherent path P . By the de�nition, any two units lyingon P are separated by an S-mincut. However, this cut fails to be transversal,since its edge-set has at least two edges in common with P . Thus, (i) followsfrom (ii).(ii) First, let us observe that a coherent path in Ffa;b;cg that entersV x cannot leave it. Indeed, the 2-partition (V x; �V x) can be representedas Cx;xy \ Cx;xz (see the proof of 3-Star Lemma), and is thus a minimum(x; fy; zg)-cut. Therefore, a coherent path in Ffa;b;cg that enters and leavesV x yields in Wx;fy;zg a coherent path that intersects at least twice the edge-set of the transversal cut (V x; �V x), a contradiction.Let now C be an S-mincut contradicting the claim, and let P be a coherentpath intersecting EC at least twice. Assume w.l.o.g. that C is an (x; fy; zg)-cut, then it is represented as a transversal cut in Wx;fy;zg. If the path Plies entirely in V x [ V xy [ V xz, then it is represented as a coherent path inWx;fy;zg, a contradiction. Thus, by 3-Star Lemma(ii), P must enter eitherV ?, or V y , or V z . However, V ? is a terminal, and a path entering V y or V zcannot leave them. Hence, in this case the contraction takes P to a coherentpath in Wx;fy;zg, and we are done. �Therefore, an arbitrary undirected graph with three distinguished verticesin the symmetric case, as well as in the asymmetric one, de�nes canonicallyan acyclic balanced locally orientable graph Ffa;b;cg. Recall that a strip canbe considered as a general graph model of a two-ended object of a constantwidth. In a sense, the locally orientable graph Ffa;b;cg with the structuregiven by Fig. 4.5b represents a general three-ended object of a constant width.The following statement stems from the proof of Lemma 4.5.Fact 4.6. Any intercell coherent path in Ffa;b;cg belongs to one of the fol-lowing four types: V x-V xy; V x-V xy-V y ; V x-V y ; V x-V ?.According to Lemma 4.1(i) and (ii), in the asymmetric subcase one canreconstruct the strip Wa;c starting from Ffa;b;cg and using the cuts (Lb; �Lb)and (Rb; �Rb). In a similar way, in the symmetric subcase one can obtaineach of the three strips Wx;y by a contraction of Ffa;b;cg de�ned in terms ofthe cuts Cx.Lemma 4.7. Let us contract Vy\Vz in Ffa;b;cg and assign to the contractedunit the 2-partition of its star into the edges crossing the cuts Cy and Cz ,respectively. The result is the strip Wy;z.Proof (to visualize the statement and the proof observe that Vy \ Vz =V ? [ V x, see Fig. 4.5). Let us denote by W the locally orientable graphde�ned in the statement of the lemma (it is well de�ned due to 3-StarLemma(ii)). We consider the two following ways of constructing the stripTec

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24Wfx;zg;y. First, we take the strip Wy;z and, according to Lemma 3.8, con-tract the cone of x (more exactly, of the unit Ux containing x) that containsz. Second, we take Ffa;b;cg and contract the inner part Vy of the cut Cy(see Lemma 4.4(i)). Since both ways yield the same result, we concludethat all units U 2 V y [ V yx [ V yz in Wy;z and Ffa;b;cg coincide, as well asthe 2-partitions at these units. Proceeding in the same way with the stripWfx;yg;z, we get the coincidence for all units, except for Ux in Wy;z andVy \ Vz that is a single unit in W ; thus, Ux = Vy \ Vz. Observe that, onone hand, the two sides of Ux in W are just the intersections of its star withthe edge-sets of the cuts Cy and Cz . On the other hand, by Fact 3.6(iii),these two cuts correspond to the cones of Ux in Wy;z via our contractionmapping. Thus, the 2-partitions at Ux in the two graphs coincide as well,and the proof is completed. �Similarly to the asymmetric case, one can build a locally orientable graphWxy, x; y 2 fa; b; cg, x 6= y, by deleting from Ffa;b;cg all the vertices ofV ? [ V z [ V xz [ V yz , z 6= x; y, and contracting V x and V y to terminals.(Informally, Wab, Wbc, and Wac are the common parts of all three stripsWa;b, Wb;c, and Wa;c). An S-mincut C of G that separates V x from V y ,and thus generates a cut C 0 in Wxy separating its terminals, is called anextension of C 0. The following statement is an analog of Lemma 4.2.Lemma 4.8.(i) The graphs Wxy are strips of width �xy 6 12�S.(ii) All their transversal cuts, and only they, are extendable to S-mincuts.(iii) Each pair of transversal cuts of Wxy andWxz , z 6= x; y, has a mutualextension to a minimum (x; fy; zg)-cut.(iv) If a 2-partition of V n V xy enters an extension of some transversalcut of Wxy, then it enters some extension of any such cut. (Informally : alltransversal cuts of this strip are interchangeable in S-mincuts).Proof. (i) Observe that, according to Lemma 4.7, one can construct Wxyin a similar way starting from Wx;y . Therefore, the proof of Lemma 4.2(i)applies with minor changes: the assumption �xy > 12�S implies the existenceof a 2-partition of cardinality less than �S separating z from fx; yg.(ii) The \all" part of the assertion is proved exactly as in Lemma 4.2. Toprove the \only" part we consider an arbitrary cut C separating V x from V y .Assume, w.l.o.g., that C is an (x; fy; zg)-cut containing x inside. ReplacingC by C [ Cy we arrive to the case considered in the proof of Lemma 4.2(ii).(iii) It follows easily from the proof of Lemma 4.2(ii) that any transversalcut C 01 ofWxy can be extended to an (x; fy; zg)-cut C1 of Ffa;b;cg. Then ~C1 =C1 \ Cz is a minimum (x; z)-cut, as the intersection of minimum (x; z)-cuts.Observe that ~C1 is an (x; fy; zg)-cut. Similarly, starting from a transversalcut C02 ofWxz we �nd an (x; fy; zg)-cut ~C2. Their union is evidently a mutualextension of C 01 and C02. Observe that ~C1 [ ~C2 is the unique mutual extensionTec

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25of C 01 and C 02, since by de�nition, each such extension contains V x inside andV ? [ V y [ V z [ V yz outside.(iv) Let C be an extension of some transversal cut of Wxy and C01 bean arbitrary transversal cut of Wxy. Assume w.l.o.g. that both x and V ?lie inside C. Recall that Cy;xy de�ned in the proof of 3-Star Lemma is aminimum (x; y)-cut. Therefore, ~C = C \ �Cy;xy is also a minimum (x; y)-cut.On the other hand, similarly to the proof of Lemma 4.2(ii), there exists anextension C1 of C 01 such that V x[V xz lies inside C1 and V y [V yz [V ? [V zoutside it. The cut ~C [ C1 is thus an extension of C 01 (since �C is an extensionof the cut separating the unit Ux in Wxy), and it partitions V n V xy in thesame way as C. �Let us analyze the incremental dynamics of FS in the symmetric subcase.Since FS is glued from the strips Wx;fy;zg (or their extensions Wx;y), it suf-�ces to study the dynamics of these strips. According to Sect. 4.1, insertionof an edge (u1; u2) modi�es each of the strips in such a way that all theunits, as well as the canonical 2-partitions of their stars, remain the same,except for the units contracted into the new one (containing both u1 andu2). Hence, by Lemma 4.4, the same holds for the locally orientable graphFS . According to the same lemma, if for some strip cWx;fy;zg the single newunit is not the terminal containing y and z, then it coincides with the newunit Unew of FS . The same is true if for some strip cWx;y the new unit is notthe unit containing z.In the description of incremental dynamics of FS we make use of thepartition into the cells V x, V xy , V ?. Therefore, to be self-contained, wehave to describe the changes in the cells; note that it su�ces for this to clearup the dynamics of the sets Vx.Below we consider all the nine distinct possibilities to insert a new edge(u1; u2), u1 2 U1, u2 2 U2, w.r.t. the cell partition.Case 1. If U1 = U2 (in particular, U1 = U2 = V ?), then FS evidentlyremains the same.Case 2. If U1 and U2 belong to the same set V xy (see Fig. 4.7a), wecan trace the dynamics in any of Wx;fy;zg, Wfx;zg;y, and Wx;y . Clearly,the de�nition of the contracted unit Unew (see Sect. 4.1) can be translatedstraightforwardly into terms of FS : Unew results from the contraction ofU1, U2, and the units and edges on the coherent paths between U1 and U2(all of them belong to V xy). Since the contracted unit in the modi�ed stripis non-terminal, this contraction is the only change in FS . Note that thechanges are localized in V xy and can be traced also in Wxy.Case 3. Similarly, if U1; U2 2 V x (see Fig. 4.7b), we can trace the changesin any of the strips Wx;fy;zg, Wx;y , orWx;z to de�ne Unew in the same way.It is easy to show that the changes are localized in V x.In all the above cases all the cuts Ca, Cb, Cc remain the same, and thusthe partition into cells is stable.Tec

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26xV x

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b)a)Fig. 4.7. Dynamics of Ffa;b;cg in the symmetric case, ICase 4. Let now U1 2 V x, U2 2 V xy (see Fig. 4.8a). As in the previoustwo cases, the new unit Unew can be de�ned via Wx;fy;zg, Wx;y , or Wx;z.The strip Wx;y can be used also for establishing the dynamics of the setVy via Lemma 4.3; in terms of this lemma, it is the set Lz . By the samelemma, the new part of bVy is V 0 = Rx(U2) n Vy. Another way is to useWfx;zg;y, where the set Vy forms the terminal Uxz containing x and z; then,the new part of bVy = bUyz is Rx(U2)nUyz . It follows from Fact 4.6 that V 0 iscontained in V xy and coincides with the cone of U2 in Wxy in the directionto the terminal containing x, minus this terminal. From the dynamics ofWx;fy;zg and Wz;fx;yg, respectively, it is clear that the sets Vx and Vz donot change. Therefore, all the cells are stable, except for bV x = V x [ V 0 andbV xy = V xy n V 0.Case 5. If U1 2 V xz , U2 2 V xy (see Fig. 4.8b), we can use stripsWx;fy;zg,Wx;y and Wx;z for determining Unew. In this case, there is no coherentpaths between U1 and U2, hence, Unew = fU1; U2g. Using strips Wx;y andWx;z, respectively, in the same way as Wfx;zg;y in the previous case, weestablish that bVy = Vy [ Rx(U2) and bVz = Vz [ Rx(U1). Therefore, bV x =V x [Rx(U1) [Rx(U2), while V xy and V xz are cut down correspondingly.V

x xVxV xyVVx xy

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xxV

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a) b)Fig. 4.8. Dynamics of Ffa;b;cg in the symmetric case, IICase 6. If U1 = V ?, U2 2 V xy (see Fig. 4.9a), we use the strip Wx;y.Once again Unew = fU1; U2g. Both Vx and Vy grow to Vx [ Rx(U2) andVy [ Ry(U2), respectively. Therefore, bV x = V x [ Rx(U2) n U2, bV x = V x [Rx(U2) n U2 and bV ? = V ? [ U2.Case 7. If U1 2 V x, U2 = V ? (see Fig. 4.9b), we consider the stripWx;fy;zg with terminals Ux, x 2 Ux, and Uyz = Vx, y; z 2 Uyz , V ? � Uyz .By Sect. 4.1, the cone Ryz(U1) is added to Uyz , resulting in bVx = Vx [Ryz(U1). Evidently, strips Wfx;zg;y and Wfx;yg;z do not change, hence VyTec

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27and Vz are stable. Using the sets bVx, bVy = Vy and bVz = Vz in accordanceto the de�nition of a cell, we get U = bV ? = V ? [ (V x \ Ryz(U1)), bV y =V y [ (V xy \ Ryz(U1)), bV z = V z [ (V xz \ Ryz(U1)), and each one of V x,V xy, V xz is lessened by the corresponding part of Ryz(U1).A special subcase occurs when U1 = Ux. Then Wx;fy;zg vanishes, andwe arrive at the asymmetric case for Ffy;x;zg. In terms of Sect. 4.2, bUx =Vx [ V ?, bLx = Vz, bRx = Vy , bVyz = Vyz and cWyz =Wyz .V z

yVV

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a) b)Fig. 4.9. Dynamics of Ffa;b;cg in the symmetric case, IIICase 8. The case U1 2 V x, U2 2 V yz (see Fig. 4.10a) is equivalent to acomposition of the two previous cases: apply the transformation of Case 6and the transformation of Case 7 with x replaced by y.To see this, assume �rst that V ? 6= ? and choose an arbitrary vertexw 2 V ?. Then the transformation of Ffa;b;cg equals the sum of the trans-formations caused by adding of two edges: (u1; w) and (w; u2). Indeed,addition of each one of these edges a�ects only one of the strips Wyz andWx;fy;zg, and their dynamics are exactly the same as in the Cases 6 and 7,respectively. The case V ? = ? can be reduced to the previous one by thefollowing trick. We add to G a new vertex w and three new edges (a; w),(b; w), (c; w). It is easy to see that S-mincuts of the new graph G� cor-respond bijectively to S-mincuts of G; more precisely, (X; �X) correspondsto (X; �X [ fwg) if and only if jX \ Sj = 1. Hence, all the cells of F�fa;b;cgcoincide with the corresponding cells of Ffa;b;cg, except for (V �)? = fwg.Since (V �)? is now nonvoid, the transformation taking F�fa;b;cg to bF�fa;b;cg isexactly as described above; to obtain bFfa;b;cg it su�ces just to delete w.Case 9. The last case occurs when U1 2 V x, U2 2 V y (see Fig. 4.10b). Itcan be considered as a composition of two Cases 7: apply the transformationof Case 7 as described and the same transformation with x replaced by y.In this situation the two transformations are not independent: their jointaction forces the set V 0 = V xy \ Ryz(U1) \ Rxz(U2) to be added to bothVx and Vy and thus to move from V xy to V ?. The validity of the abovedescription can be observed in the same way as in the previous case; the twoedges added in this case are the edges between U1 and V ? and between V ?and U2.Tec

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28The only special subcase not yielded by double application of Case 7occurs if U1 = Ux, U2 = Uy . It results in vanishing of both strips Wx;fy;zgandWy;fx;zg, and thus leads to degeneration of Ffa;b;cg to cWfx;yg;z = bFx;z =bFy;z ; evidently, cWfx;yg;z coincides with Wfx;yg;z.V

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Fig. 4.10. Dynamics of Ffa;b;cg in the symmetric case, IV4.4. An arbitrary subset S with the path structure. Comparing theasymmetric and the symmetric cases considered, one can assign to the �rstof them the structure of the path (a; b; c), and to the second, of the 3-starwith terminals a; b; c (Fig. 4.11a,b). The 1-cuts of these \skeleton" structurescorrespond bijectively to distinct partitions of fa; b; cg by minimum fa; b; cg-cuts in the original graph.a

b

a b ccba b bc 2

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1 r-1

a) b)Fig. 4.11. Skeleton structuresLet S = fa = b0; b1; : : : ; br�1; br = cg; assume that there exist ex-actly r distinct partitions of S by S-mincuts and each of them is of theform (fa; b1; : : : ; bi�1g; fbi; : : : ; br�1; cg), 1 6 i 6 r. This means that theskeleton structure (in the sense de�ned above) for this case is the path(a; b1; : : : ; br�1; c) (see Fig. 4.11c). Similarly to the case of the path (a; b; c)(Sect. 4.2), all S-mincuts are represented by the (a; c)-stripWa;c. We de�nethe structure of local orientability on the graph FS by keeping that of Wa;cat all the units, except for the units Ubi containing bi, 1 6 i 6 r � 1, whichare now regarded as terminals. As in Sect. 4.2, FS is acyclic and balancedand all S-mincuts of G are transversal cuts in FS .For each i, 0 6 i 6 r�1, let (Li; �Li) be the fa; b1; : : : ; big-tight (fa; b1; : : : ;big; fbi+1; : : : ; br�1; cg)-mincut. Similarly, for each i, 1 6 i 6 r, let (Ri; �Ri)be the fc; br�1; : : : ; big-tight (fc; br�1; : : : ; big; fbi�1; : : : ; b1; ag)-mincut. Be-sides, we de�ne Lr and R0 to be the entire set of units of FS . The propertiesTec

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29of the cuts (Li; �Li) and (Ri; �Ri) are similar to those of (Lb; �Lb) and (Rb; �Rb)(see Lemma 4.1) and of Cx (see Lemma 4.4(i) and Lemma 4.7). We omit theproof, since it is similar to that of Lemma 4.1; for illustration see Fig. 4.12.Lemma 4.9.(i) For any pair i; j, 0 6 i < j 6 r, one has Li � Lj and Ri � Rj.(ii) For any i, 0 6 i 6 r, Li and Ri are the reachability cones of the unitUbi in Wa;c, and hence Li \Ri coincides with this unit.(iii) The sides of the star of Ubi in Wa;c are induced in FS by edge-setsof (Li; �Li) and (Ri; �Ri).(iv) For any pair i; j, 0 6 i < j 6 r, the contraction of Li and Rj in FSinto two terminals gives Ffbi;:::;bjg.a b b ci j

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throughFig. 4.12. Cell structure for the path caseBy Lemma 4.9(iii), the strip Wa;c can be restored from FS . Observethat the sets fbi; : : : ; bjg, i < j, have the path structure similar to that of S.Therefore, by Lemma 4.9(iii) and (iv), the stripsWbi;bj can be restored fromFS as well. A unit which is not contracted to a terminal in the constructiondescribed in Lemma 4.9(iv) is said to participate in Wbi;bj .The two nested families fLig and fRig de�ne a cell structure on the setof units. Namely, the cell Qij , 0 6 i 6 j 6 r, is de�ned as the intersection(LjnLj�1)\(RinRi+1), where L�1 = Rr+1 = ?. Observe that Qii is exactlythe unit containing bi. Part (iv) of Lemma 4.9 implies that the units of thesame cell participate in the same \partial" strips Wbp;bq . More exactly, thefollowing statement holds.Lemma 4.10. The units of the cell Qij participate in the strip Wbp;bq ifand only if the paths (bi; : : : ; bj) and (bp; : : : ; bq) in the skeleton structurehave at least one common edge.Proof. According to (iv) above, to obtain Wbp;bq we contract the cells Qijwith j 6 p or i > q, and the assertion follows. �Tec

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30Edges between cells and, generally, mutual reachability of their units, aresubject to the following restriction.Lemma 4.11. Let U1 and U2 be mutually reachable in FS , Ul 2 Qil;jl ,l = 1; 2, and i1 < i2. Then:(i) j1 6 j2;(ii) if U1 is a stretched unit and U3 lies in the same reachability cone ofU1 as U2, U3 2 Qi3;j3 , then i1 6 i3 and j1 6 j3.Proof. (i) Assume that i1 < i2, j2 < j1, and there exists a coherent pathP2 between U1 and U2. Evidently, U1 is not a terminal. Let us extend P2behind U1 up to a terminal Ubp . Without loss of generality we may assumethat p > i1 (the case p < j1 is treated in the same way). Let us considerthe S-mincut (X; �X) with X = Lj1 \ �Ri1+1. Evidently, the endpoints U2and Ubp of the extended path lie outside this cut, while an intermediate unitU1 lies inside this cut. So, the extended path intersects the transversal cut(X; �X) at least twice, a contradiction.(ii) Let Ubp be as above. From (i) we get immediately that either p > j1,or p 6 i1. If p > j1, then we set i = minfi2; j1g; now both Ubp and U2 belongto Ri, while U1 does not (since i1 < i2), a contradiction. Hence, p 6 i1.Let us now extend a coherent path P3 between U1 and U3 behind U3 upto a terminal Ubq (since U3 is not necessarily a stretched unit, it may happenthat Ubq = U3). Observe that P3 can be extended up to Ubp ; we denote theobtained coherent path by P 03. Once more we get from (i) that either q > j1,or q 6 ii. If q 6 i1, then the endpoints Ubq and Ubp of P 03 belong to Li1 ,while its intermediate unit U1 does not, a contradiction. Hence, q > j1.Assume now that i3 < i1; in this case U3 6= Ubq . Let us consider thepart of P 03 between U1 and Ubq . Both its endpoints belong to Ri1 , while itsintermediate unit U3 does not, a contradiction.Assume now that i3 = i1 and j3 < j1; again we have U3 6= Ubq . Let usconsider the same path as in the previous case. Both its endpoints do notbelong to Lj3 , while its intermediate unit U3 does, a contradiction.Therefore, i1 = i3 implies j1 6 j3. Finally, i1 < i3 implies the sameinequality. �Remark. Observe that the �rst assertion of the above Lemma can be re-garded as a generalization of Lemma 2.1(i).Let us now extend the de�nition of the cell structure to the edges ofFS . Let e = (U1; U2) be an edge of FS , and assume that U1 2 Qi1j1 andU2 2 Qi2j2 . We put i = minfi1; i2g, j = maxfj1; j2g, and say that e belongsto the cell Qij . Observe that by Lemma 4.11 there are three types of edgesbelonging to a cell Qij : internal (both endpoints belong to Qij), external(one endpoint belongs to Qij and the other one either to Qij0 , i 6 j 0 < j, orto Qi0j , i < i0 6 j), and through (one endpoint belongs to Qij0 , i 6 j0 < j,and the other one to Qi0j , i < i0 6 j); for an illustration see Fig. 4.10.Tec

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31This de�nition can be justi�ed as follows. Insert a dummy vertex u intothe edge e, thus dividing it into e1 and e2. Evidently, if an S-mincut (X; �X)of the initial graph is crossed by e, then upon insertion of u it is replacedby two S-mincuts of the modi�ed graph: (X [ fug; �X) and (X; �X [ fug).Conversely, let (Y; �Y ), u 2 �Y , be an S-mincut of the modi�ed graph crossed,say, by e1. Then (Y [ fug; �Y n fug) is another S-mincut of the modi�edgraph, and it is crossed by e2. Finally, if an S-mincut is not crossed by e,then it remains intact in the modi�ed graph, and vice versa. It follows fromthe above discussion that the only di�erence between the initial FS and themodi�ed one is the new unit U that consists of the single dummy vertex uand belongs to the cell Qij ; U , together with the incident edges e1 = (U1; U)and e2 = (U2; U), replaces the edge e in FS . This dummy unit, in a sense,represents the edge e, and thus justi�es the above de�nition. Observe thatthis de�nition is stable w.r.t. insertion of dummy vertices. Namely, if ebelongs to the cell Qij , then both e1 and e2 belong to the same cell.e1

2e

e1

ijQ ijQ

W ij

Fig. 4.13. To the definition of the strip W ijFor any pair i; j, 0 6 i 6 j 6 r, excluding cases i = j = 0 and i = j = r,we build from FS a locally orientable graphW ij with the set of nonterminalsQij (see Fig. 4.13). We delete SfQpq : p < i 6 j < qg and SfQpq : i < p 6q < jg (observe that by Lemma 2.1 there are no edges between Qij and thedeleted sets). Next, we contract the setsSfQpq : p 6 i; q 6 j; (p; q) 6= (i; j)gand SfQpq : p > i; q > j; (p; q) 6= (i; j)g into two terminals a0 and c0.Finally, we delete the edges between a0 and c0 that do not belong to Qij .Observe that this de�nition is consistent with the de�nition of Wac givenin Sect. 4.2 (W02, in our new notation), because all the edges between a0and c0 in Ffa;b;cg belong to Q02. Evidently, all the edges that belong to Qijparticipate inW ij. Besides, each external edge participates in one more suchgraph, and each through edge in two more such graphs. Extensions of cutsof W ij to S-mincuts are de�ned similarly to the case analyzed in Sect. 4.2.The following statement is an analog of Lemmas 4.2 and 4.8.Lemma 4.12.Tec

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32(i) The graph W ij is a strip of width �ij; if j = i then �ij > �S=2, whileif j > i+ 2 then �ij < �S=2.(ii) All its transversal cuts, and only they, are extendable to S-mincuts.(iii) For any sequence i1 < i2 < � � � < il 6 jl < � � � < j2 < j1, any setof l transversal cuts of W i1j1 ; : : : ;W iljl has a mutual extension to an S-cut.Moreover, if jl > il + 2 then �i1j1 + � � �+ �iljl < �S=2.(iv) Let i1 < i2 < � � � < il 6 jl < � � � < j2 < j1 and let a 2-partitionof V n fQi1j1 [ � � � [ Qiljlg enter a mutual extension of a set of transversalcuts of W i1j1 ; : : : ;W iljl . Then it enters some mutual extension of any setof such cuts. (Informally : all sets of transversal cuts for such a sequence ofstrips are interchangeable in S-mincuts).Proof. (i) The fact that W ij is a strip is proved similarly to Lemma 4.2(i)taking into account that the sets Xa = SfQpq : p 6 i; q 6 j; (p; q) 6= (i; j)gand �Xc = SfQpq : p > i; q > j; (p; q) 6= (i; j)g de�ne transversal cutsCa = (Xa; �Xa) and Cc = (Xc; �Xc) (shown by a double line on Fig. 4.13).The inequality for the case j = i follows from the fact that the cardinalityof the cut (Ubi ; V n Ubi) exceeds �S . In the case j > i + 2 we see that allthe edges of FS incident to a0-type terminals in the stripsW ij andW i+1;i+1belong to the edge-set of Ca. Therefore, �ij + �i+1;i+1 6 c(Ca) = �S . Since�i+1;i+1 > �S=2, we are done.(ii) Let Xa and �Xc be de�ned as above. Given any transversal cut C 0 =(a0 [ Z; �Z [ c0) of W ij, let us consider the edge-set of the 2-partition C =(Xa [ Z; �Z [ �Xa) of V . As compared with the edge-set of Ca, it looses theedges between Xa and Z and gets the edges between Z and �Z[ �Xc. However,the same holds for the cut C 0 as compared with the cut ofW ij that separatesa0. Since, by (i), the cuts in the second pair have equal cardinalities, thesame is true for the �rst pair. Thus C is a minimum S-cut.Conversely, let (X; �X) be an S-mincut separating Xa from �Xc. By thesame reasons as above, it generates a 2-partition of cardinality �ij in W ijthat is a minimum (i.e., transversal) cut of W ij .(iii) Let C0k be a transversal cut of W ikjk , 1 6 k 6 l. Denote by Ck theextension of C 0k de�ned as in the proof of (ii) and consider the S-mincutC = C1 [ � � � [ Cl. Inequalities i1 < i2 < � � � < il 6 jl < � � � < j2 < j1guarantee that C is an extension for any of the C0k, and hence, is a mutualextension of the whole set fC01; : : : ; C 0kg. The same inequalities guaranteethat no edge is counted more than once in the sum �i1j1 + � � �+ �iljl ; therest of the proof of the inequality �i1j1 + � � �+�iljl < �S=2 follows the proofof (i).(iv) Assume that l = 1. Let C0 be an extension of some transversal cutof W i1j1 and C 0 be an arbitrary transversal cut of W i1j1 . Take �Xc as aboveand put X�c = Xc n Qi1j1 ; then C� = (X�c ; �X�c ) is an S-mincut. Now de�nethe cut C of G as in the proof of (ii); then (C0 \ C�) [ C is an extension ofC 0 that divides V nQi1j1 in the same way as C0. For l > 1 the proof appliesTec

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33with evident minor changes. �Dynamics of FS , S = fa = b0; : : : ; br = cg, in the case of the pathstructure are induced by those of Ffa;cg = Wa;c, in the same way as it wasdone for Ffa;b;cg in the asymmetric subcase (see Sect. 4.2). To be moreprecise, let U1 2 Qi1j1 , U2 2 Qi2j2 , i1 6 i2. Elements of S \fall inside" Unewwhen j1 6 i2; in this case Unew is a terminal and it contains all the bk suchthat j1 6 k 6 i2 (see Fig. 4.14a).Concerning the dynamics of the sets Lk (those ofRk are similar), Lemma 4.3can be generalized as follows.Lemma 4.13. The sets Lk, i1 6 k < i2, and only them are extended (atleast by U2). The new part of bLk is the union of the units U 0 2 Qi0j0 suchthat i0 > k and U 0 2 Ra(U2).The proof of Lemma 4.13 is similar to that of Lemma 4.3 (see Fig. 4.14b).2U

LkLk

1U

bj1

....................

1U2U

a) b)

bi2Fig. 4.14. Dynamics in the case of the path structureIn the case when Unew is not a terminal, its new cell can be de�ned viathe characterization of the sets bLk, bRl given in Lemma 4.13.Corollary 4.14. Let imin = minfi1; i2g, imax = maxfi1; i2g, jmin = minfj1;j2g, jmax = maxfj1; j2g. The new unit Unew belongs to the cell bQiminjmax.Let a unit U 0 2 Qij move to the cell bQi�j�.(i) If i > imin and U 0 2 Ra(Uimax) then i� = imin, otherwise i� = i.(ii) If j < jmax and U 0 2 Rc(Ujmin) then j� = jmax, otherwise j� = j.5. The flesh and the skeleton of the connectivity carcass5.1. The system of units, or the esh of the connectivity car-cass. Let us consider an arbitrary bunch B of S-cuts, and let (SB; �SB) be thecorresponding partition of S. In fact, B is the set of all minimum (SB; �SB)-cuts; thus the structure of B can be represented by the strip WSB; �SB . Forbrevity, in what follows this strip is denoted by WB, the corresponding quo-tient mapping by �B , and the units ofWB are called B-units. Therefore, thefollowing statement holds.Fact 5.1. Any bunch of S-cuts B is represented by the strip WB and themapping �B so that the �B-inducing provides a bijection between the transver-sal cuts of WB and the S-cuts in B.Tec

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34Evidently, for the bunch �B of cuts opposite to cuts in B, the strip W �Bcoincides with WB.By the de�nition, the partition of V into S-units is a re�nement of itspartition into B-units. The following statement shows that only the twoterminal B-units can be re�ned.Lemma 5.2. Each nonterminal B-unit is an S-unit.Proof. The proof is similar to that of Lemma 4.4(i). Let U be a nonterminalB-unit subdivided by an S-mincut C. Evidently, there exist vertices x 2 SBand y 2 �SB separated by C; assume w.l.o.g. that x lies inside C. By thede�nition of B-units, the bunch B contains cuts C1 and C2 separating SBfrom U and U from �SB, respectively. Observe that all the three cuts C, C1,and C2 are minimum (x; y)-cuts. Let us consider the cut C1 [ (C \ C2). Itcontains SB inside, contains �SB outside, and thus belongs to B. However, itsubdivides U , a contradiction. �By the de�nition, the star of any nonterminal vertex of the locally ori-entable graph WB is partitioned into two sets of equal cardinality. Thus, byLemma 5.2, the star of the corresponding vertex of FS is halved. The crucialobservation is that the arising 2-partition does not depend on the choice ofB (see Lemma 4.4(iii) for a particular case of the same statement).Lemma 5.3. If an S-unit participates in several bunch strips as a nonter-minal unit, then the 2-partitions of its star in all these strips coincide.Proof. Let U be an S-unit participating in bunches B and B0. Without lossof generality, the 2-partition (SB0 ; �SB0) divides SB. Therefore, for any x 2 �SBone can choose y 2 SB separated from x by this 2-partition. Similarly tothe proof of Lemma 4.4(iii), both stripsWB and WB0 are contractions of thesame strip Wx;y (in this case Lemma 3.8 is used repeatedly). Hence, the 2-partitions at U in these strips are inherited fromWx;y and thus coincide. �1e

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35A unit that participates at least in one bunch strip is called stretched ; thecanonical partition of its star indicated in Lemma 5.3 is called inherent . Allthe other units are called terminal. A unit is called heavy if it intersects S.Observe that only terminal units (but, in general, not all of them) can beheavy. To the star of a terminal unit of FS we assign the trivial inherent2-partition. Thus, the quotient graph FS acquires a canonical structure of alocally orientable graph; it is said to be the esh of the connectivity carcassof S (see Fig. 5.1a,b). Below we prove several basic properties of FS .According to the de�nitions above, we build FS by \glueing together" allthe bunch strips. Conversely, each bunch strip can be reconstructed fromFS with the help of contractions. Indeed, let B be an arbitrary bunch ofS-mincuts. The intersection of all cuts in B is called the tight cut of B; theunion of all cuts in B is called the loose cut of B (evidently, each of themseparates a terminal of WB from all other units). Observe that the tightcut of B is just the SB-tight (SB; �SB)-mincut, while the loose cut of B isthe opposite to the �SB-tight ( �SB; SB)-mincut. The following statement is ageneralization of Lemma 4.1(iii) and Lemma 4.4(i).Lemma 5.4. For any bunch B, the strip WB is obtained from FS by con-tracting all the units inside the tight cut of B and all the units outside theloose cut of B into terminals.Proof. Follows immediately from Lemmas 5.2 and 5.3 (see the proof ofLemma 4.4(i) for details). �Parts (i) and (ii) of the next statement provide a generalization of Lemma 4.5.Theorem 5.5.(i) The esh FS is a balanced acyclic locally orientable graph.(ii) Each S-mincut is transversal in FS.(iii) Let ~n and ~m be the numbers of vertices and edges of FS , respectively.Then ~n 6 n, ~m 6 m, ~m = O(�S~n).Proof. (i) The fact that FS is balanced is clear, since inherent partitionsat stretched units of FS come from those in bunch strips (see Lemma 5.3).Acyclicity of FS follows from part (ii) of this theorem exactly in the sameway as in the proof of Lemma 4.5.(ii) Assume to the contrary that the claim is violated by some pair G; Sand the corresponding esh FS . Then there exists a minimal counterex-ample, namely, an S-mincut C and a coherent path P in FS such that Pintersects the edge set of C at least twice and the number of units in P isminimal among all such pairs C; P (see Fig. 5.2). By minimality, the end-points of P lie outside C, and all the inner units of P lie inside C (up to ipping of C). Let U be an arbitrary inner unit of P ; since P is a coherentpath, U is a stretched unit. Therefore, there exists a bunch in which Uparticipates. Let C1 be the S-mincut in this bunch de�ned by a cone R of Uin the corresponding strip and C2 be the S-mincut in this bunch de�ned byTec

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36RnU . Up to taking the other cone of U , we may assume that both pairs C1,C and C2, C satisfy the conditions of Lemma 2.1. Thus, by Fact 2.2, bothC1 \ C and C2 \ C are S-mincuts; moreover, they belong to the same bunchand their inner parts di�er exactly by U .C1

P

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C2

CC2

Fig. 5.2. To the proof of Theorem 5.5Let us traverse P from an endpoint up to U and take the edge enteringU and the next edge of P . By Fact 3.3, exactly one of the above edgesintersects the edge set of C2 \ C; let U 0 be the other endpoint of this edge.Since U 0 lies inside C2 \ C, it lies inside C as well, and thus cannot be anendpoint of P . Let P 0 be the subpath of P leaving U by the edge (U; U 0) andending at the corresponding endpoint of P . Since both U and this endpointof P lie outside C2 \ C, the pair C2 \ C; P 0 contradicts the minimality of thepair C; P .(iii) The �rst two inequalities are evident. Let us show that ~m = �S(~n�1).Since any two units are separated by an S-mincut, the size of a minimumcut between them in FS is at most �S . Let us consider the Gomory-Hutree of FS (see [GH]). Recall that its vertices correspond bijectively to theunits, and thus any of its edges de�nes a cut of FS . By [GH], this cut is aminimum cut between some two units, since its size is at most �S ; there are~n � 1 such cuts. Since each edge of FS takes part in at least one such cut,their total number ~m does not exceed �S(~n� 1), as required. �The esh can be considered, in a sense, as a generalization of a strip.By the way, at early stages we called our structure \branching dag". Themeaning of such a nickname will become more clear in the next subsection.5.2. The system of bunches, or the skeleton of the connectivitycarcass: odd case. For the case S = V , the structure of all S-mincuts isrepresented by a cactus tree (see [DKL]); it is, in fact, a tree if the graph con-nectivity is odd. The construction is based on Crossing Lemma [Bi, DKL].A generalization of this lemma to S-mincuts and the existence of a cactustree representation of all 2-partitions of S by S-mincuts were stated (with-out proof) independently in [N, W93] (see also [DV1]). The generalizationof the lemma is straightforward (see below); for a proof of the existence ofTec

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37a cactus tree representation see [DN]. Below we describe this representationfor the odd case.Two S-cuts C and C 0 are said to be S-crossing if all the four subsetsSC \ SC0 , SC \ �SC0 , �SC \ SC0 , �SC \ �SC0 are nonvoid.S-Crossing Lemma. Let C and C0 be S-crossing S-mincuts, then(i) c(VC \ VC0; VC \ �VC0) = c(VC \ �VC0; �VC \ �VC0) = c( �VC \ �VC0 ; �VC \ VC0) =c( �VC \ VC0 ; VC \ VC0) = �S=2;(ii) c(VC \ VC0; �VC \ �VC0) = c(VC \ �VC0; �VC \ VC0) = 0.Hence, the 2-partitions C \C 0, C \ �C 0, �C \ �C 0, �C \C0 are S-mincuts as well.The proof coincides with that of Crossing Lemma, with an additionalobservation that in our case all 2-partitions involved are actually S-mincuts.It follows immediately from S-Crossing Lemma(i) that for �S odd thereare no S-crossing S-mincuts. Whenever this S-crossing-free property holds,we can represent the system of bunches of S-mincuts by a tree (for an exam-ple see Fig. 5.1c). Let ~� 6 � be the number of (�S + 1)-connectivity classesin S.Theorem 5.6. For any S � V with S-crossing-free property there exist aunique tree HS with the node set NS and a mapping 'S : S ! NS such thatjNS j = O(~�) and 'S-inducing provides a bijection between the cuts of HSand the pairs of mutually opposite bunches of S-mincuts.Proof. It is an immediate corollary of a more general statement concerningtree models for crossing-free systems of 2-partitions of an arbitrary set (see[DW, Sect. 4.1], [DN]). �The tree HS is said to be the skeleton of the connectivity carcass of theset S. Its edges are called structural edges (to distinguish them from theedges of the graph G). Any cut of HS is de�ned by a single structural edge.Observe that the image 'S(S) does not necessarily coincide with NS . Thenodes N for which '�1S (N) = ? are called empty nodes (e.g., node N onFig. 5.1c). It follows from the construction that the degree of each emptynode is at least three. The preimages of the nonempty nodes in NS areexactly the classes of (�S + 1)-connectivity of S.By Theorem 5.6, for any S with �S odd the system of bunches of S-mincuts is represented by the skeleton HS , which is a tree. For �S even thismay be also the case, if the S-crossing-free property holds. In general, ananalog of Theorem 5.6 is valid, in which the skeleton HS is a cactus tree. Itmeans that each pair of mutually opposite bunches is represented by a cut ofHS , which is de�ned either by a structural edge not belonging to any cycle,or by a pair of structural edges belonging to the same cycle. For details see[DV1], [DV3].In the rest of the paper we assume that the skeleton is a tree, thus com-pletely covering the odd case.Tec

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386. Structure and properties of the connectivity carcass6.1. Tight cuts. Let " be an arbitrary structural edge of HS and N beone of its endpoints. By Theorem 5.6, this edge de�nes two mutually oppo-site bunches of S-mincuts; these two bunches are represented by a strip (seeFact 5.1), which we denoteW("). (For example, Figs. 5.1d,e show the stripsW(N;N1) andW(N;N3), respectively.) Consider the cut of HS correspond-ing to " and containing N inside. The intersection of all S-mincuts of thebunch represented by this cut of HS is the tight cut of this bunch. This cutis said to be the N -tight cut in direction " and is denoted by C(N; ") (e.g.,the cut C3 on Fig. 5.1a is the N -tight cut in direction (N;N1)).A path in HS is an alternating sequence of distinct nodes and structuraledges; we consider the opposite sequence as the same path traversed in theopposite direction. The unique path between two arbitrary nodes M , N ofthe skeleton is denoted by [M;N ]. For any two structural edges "0 and "00there exists a unique path that starts by one of them and ends by the otherone; we denote this path by ["0; "00].It turns out that tight cuts are monotonous along paths on the skeleton.More precise, the following generalization of Lemma 4.9(i) holds.Lemma 6.1. For any path P = (N0; "1; N1; : : : ; "r; Nr) on the skeleton HS ,r > 2, one has C(N0; "1) � C(Nr�1; "r).Proof. Observe that the sets S \VC(N0;"1) and S \VC(Nr�1;"r) are nonempty,since the cuts in question are S-cuts. Moreover, S \ VC(N0;"1) is containedin S \ VC(Nr�1;"r), since both sets are 'S-preimages of subtrees of HS andthe subtree of the former is contained in that of the latter; the set inclusionis strict since the partitions of S by distinct bunches are distinct.Let a and b be arbitrary vertices in S \ VC(N0;"1) and S \ �VC(Nr;"r), re-spectively. Both C(N0; "1) and C(Nr�1; "r) are minimum (a; b)-cuts, so theirintersection C is a minimum (a; b)-cut as well. By the set inclusion provedabove, the cut C belongs to the bunch de�ned by "1, and thus coincides withC(N0; "1), since the latter cut is tight. This coincidence, together with thestrict inclusion S \ VC(N0;"1) � S \ VC(Nr�1;"r), implies the assertion of thelemma. �Corollary 6.2. For any path P = (N0; "1; N1; : : : ; "r; Nr) on the skeletonHS , r > 2, one has VC(N0;"1) \ VC(Nr;"r) = ? and VC(N1;"1) 6� VC(Nr�1;"r).Remark. Let "0 and "00 be two arbitrary structural edges and C 0 and C 00 betwo tight cuts in directions "0 and "00, respectively. Observe that Lemma 6.1and Corollary 6.2 cover all the possible choices for C 0 and C 00.6.2. Reconstruction of pairwise strips. According to Lemma 5.4, onecan obtain the strip representation for any bunch of S-mincuts just by acontraction of FS . This fact holds also in a more general situation, whenone is interested in the strip representation Ws1;s2 for an arbitrary pair ofTec

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39vertices s1; s2 2 S, or, more generally, in the strip WS1;S2 for an arbitrarypair of disjoint subsets S1; S2 � S, provided the cardinality of (S1; S2)-mincuts equals �S . Observe that in this case the (S1; S2)-mincuts, whichare represented by WS1;S2 , are exactly the S-mincuts separating S1 fromS2. In this subsection we describe the contractions of FS that give thecorresponding strips. The reader can use for illustration Fig. 5.1d (S1 =fs1g, S2 = fs2; s02; s3g), Fig. 5.1e (S1 = fs3g, S2 = fs1; s2g), Fig. 5.1f(S1 = fs2g, S2 = fs3g).Let T1 and T2 be two edge-disjoint subtrees of HS . The link of T1 and T2is de�ned as the unique path with one endpoint in T1 and the other one inT2 and having no edges in common with T1 and T2. We denote the link ofT1 and T2 by L(T1; T2). Observe that the link of two distinct nodes M andN is just the path [M;N ], the link of two distinct edges "0 and "00 extendedby these edges themselves gives exactly the path ["0; "00], and the link of twosubtrees intersecting by a node is this very node.Let S1 and S2 be disjoint subsets of S. By Theorem 5.6, a bunch ofS-mincuts separates S1 from S2 if and only if the deletion of the corre-sponding edge of HS produces two subtrees containing 'S(S1) and 'S(S2),respectively. Let T (S1) and T (S2) be the minimal subtrees of the tree HScontaining 'S(S1) and 'S(S2), respectively. For brevity, we write L(S1; S2)instead of L(T (S1); T (S2)). The following statement follows easily fromTheorem 5.6 and Lemma 6.1.Fact 6.3.(i) If T (S1)\T (S2) 6= ?, then there are no S-mincuts separating S1 fromS2.(ii) Otherwise, the edges of HS corresponding to the bunches of S-mincutsseparating S1 from S2 form the link L(S1; S2).(iii) Under the assumptions of (ii), let N1 be the endpoint of L(S1; S2)lying in T (S1) and "1 be the edge of L(S1; S2) incident to N1. Then theS1-tight (S1; S2)-mincut coincides with the N1-tight cut in direction "1.Now we can prove the main result of this subsection, which is a gener-alization of Lemma 4.1(i),(ii), Lemma 4.7, and Lemma 4.9 (ii)-(iv). For anillustration see Fig. 6.1.S ,S(L

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40Theorem 6.4. Let S1 and S2 be two subsets of S such that T (S1)\T (S2) =?. The strip WS1;S2 is obtained as follows.The underlying graph of WS1;S2 is obtained from FS by contractions ofcertain subsets of units to a single heavy unit. These subsets correspondbijectively to the nodes of the link L(S1; S2). The subset UN that correspondsto a node N is equal to the intersection of the inner sides of N -tight cuts indirections of the edges of L(S1; S2) incident to N .The inherent partition at any noncontracted unit remains the same. Theinherent partition at a new unit UN is induced by the edge-sets of the N -tightcuts de�ning UN .Proof. Let us delete all the edges of the path L(S1; S2). The trees of theforest thus obtained correspond bijectively to the nodes of L(S1; S2): a nodeN corresponds to the tree TN that contains N . Denote by �N the inverseimage of the set of all the nodes of TN under the mapping 'S . Observe that�N is a nonempty subset of S. Indeed, if N is an endpoint of the link, then�N is just the intersection of a side of a certain S-mincut with S, and is thusnonempty. If N is an inner node of the link, then �N = ? would imply thatthe cuts of HS de�ned by the two structural edges incident to N 'S-inducethe same 2-partition of S, in a contradiction to Theorem 5.6 (cp. the proofof Lemma 6.1). Besides, by Fact 6.3(ii), the S-mincuts separating S1 fromS2 do not divide �N . Therefore, in any case all the set �N lies in a singleheavy unit of the stripWS1;S2 , which we denote by U 0N . Observe that if N isan endpoint of the link, then U 0N is a terminal of WS1;S2 (since �N containsone of S1 and S2).Let now N1 and N2 be two neighboring nodes in the link such that N1and 'S(S1) lie on the same side of the edge (N1; N2) in HS . Let us considerthe set of all (S1 [ �N1 ; S2 [ �N2)-mincuts (all of them are represented inWS1;S2). It is easy to see that the subtrees T (S1 [ �N1) and T (S2 [ �N2)contain nodes N1 and N2, respectively. By Fact 6.3(ii), this set of cuts isrepresented by the strip W(N1; N2) de�ned by the structural edge (N1; N2).Therefore, by Lemma 3.8, to get the strip W(N1; N2), it is su�cient tocontract the corresponding cones of the nodes U 0N1 and U 0N2 in WS1;S2 .By Facts 3.6(iii) and 6.3(ii) and Lemma 6.1, these cones are just the in-ner sides of the tight cuts C(N1; (N1; N2)) and C(N2; (N1; N2)), respectively.Since the intersection of the two opposite cones of any unit in a strip coin-cides with this unit, we thus get that the units U 0N constructed in the proofcoincide with the sets UN de�ned in the formulation of the theorem.Observe that, by Lemma 5.4, one can obtainW(N1; N2) directly from FSby contracting the inner sides of the same tight cuts C(N1; (N1; N2)) andC(N2; (N1; N2)). Therefore, there is a bijection between the units of FS andWS1;S2 that lie between pairs of the above described tight cuts; moreover,this bijection preserves the inherent partitions. Let us consider the set ofunits in WS1;S2 that do not participate in that bijection. One can deduceTec

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41from Lemma 6.1 and Corollary 6.2 that each such unit U lies either insidethe tight cut C(N; (N;M)), where N is an endpoint of the link and (N;M)belongs to the link, or in the intersection of the inner sides of the two tightcuts C(N; (N;N1)) and C(N; (N;N2)), where N is an inner node of the linkand N1 and N2 are the neighbors of N in the link. In the �rst case, U is oneof the terminal units of WS1;S2 (by Fact 6.3(iii)). In the second case, U liesin the intersection of the two opposite cones of the unit UN in WS1;S2 , andthus coincides with UN . The claim concerning the inherent partition at UNfollows immediately from the above construction and Fact 3.3. �Remarks. 1. Let us distinguish a subset of coherent paths in FS by pre-venting their inner units from \falling inside" any unit UN , N 2 L(S1; S2).By the construction above, these paths correspond naturally to the coherentpaths of WS1;S2 without inner units UN , N 2 L(S1; S2).2. Observe that for any path P = (N0; "1; : : : ; "r; Nr) in HS , the link ofthe subsets of S lying inside C(N0; "1) and C(Nr; "r) is exactly P , since thereare no empty leaves in HS . The strip corresponding to these two subsets wedenote by W(P ).6.3. The projection mapping. Let U be an arbitrary unit of the eshFS . We say that U is projected to an edge " of the skeleton HS if U is anonterminal unit of W(") (for example, unit U2 on Fig. 5.1b is a stretchedunit of the strip W(N1; N) shown on Fig. 5.1d, and thus U2 is projected tothe edge (N1; N)). In other words, U is projected to " = (N1; N2) if it liesoutside the N1-tight and N2-tight cuts in direction ".Theorem 6.5.(i) For any stretched unit U of the esh, the set �S(U) of structural edgesto which U is projected is nonempty and is the edge set of a path in theskeleton.(ii) For any terminal unit U of the esh, the set of structural edges towhich U is projected is empty, and there exists a unique node �S(U) of theskeleton such that U belongs to all �S(U)-tight cuts.Proof. (i) Let us show that there exists a path on the skeleton containingall the structural edges in �S(U). If there is only one such edge, then thestatement is trivial; thus in what follows we assume that j�S(U)j > 2. Letus take a path P = (N0; "1; N1; : : : ; Nr�1; "r; Nr) that is inclusion{maximalamong all paths with both end-edges in �S(U). Suppose, to the contrary,that there exists an edge " 2 �S(U) that does not belong to P (see Fig. 6.2a).Then the link of " and P meets P in an inner node Ni (otherwise P couldbe extended). We denote by "0 the edge of this link incident to Ni. ByLemma 6.1, U lies outside the cuts C(Ni; "i), C(Ni; "i+1), and C(Ni; "0).However, this contradicts to the following generalization of 3-Star Lemma(i).Tec

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42NrN0

rεε

............................................................

a) b)

1ε

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Ni

εN

C C *Fig. 6.2. To the proofs of Theorem 6.5 and Lemma 6.6Lemma 6.6. Let N be a node of HS , "1, "2, "3 be structural edges incidentto N , Ci = C(N; "i), i 2 f1; 2; 3g. Then \i �VCi = ?.Proof. Let us choose a vertex si 2 S in such a way that si 2 �VCi , i = 1; 2; 3.Vertices si are distinct, since they belong to inverse images (under 'S) ofdisjoint subtrees of HS . Observe that T (fsj ; skg) is a path containing both"j and "k , but not containing "i. Thus, by Fact 6.3(iii), the fsj ; skg-tight(fsj ; skg; si)-mincut coincides with Ci. Therefore, Ci is exactly the cut Csias de�ned in Sect. 4.3, and the assertion of the lemma follows from 3-StarLemma(i). �Remark. Clearly, the remaining assertions of 3-Star Lemma can be general-ized to our case as well.It remains to show that each edge (Ni�1; "i; Ni) in P belongs to �S(U). In-deed, U lies outside both C(N1; "1) and C(Nr�1; "r) and thus, by Lemma 6.1,outside both C(Ni; "i) and C(Ni�1; "i), as required.(ii) Let now U be a terminal unit of the esh. Then, for each struc-tural edge ", U lies inside exactly one of the two tight cuts in direction". Let us consider the collection I of all tight cuts that contain U in-side and have inclusion-minimum inner sides. Let C0 and C 00 be two arbi-trary cuts in I, "0 and "00 be the corresponding structural edges, and let["0; "00] = (N0; "0; N1; : : : ; Nr�1; "00; Nr). Then, by Lemma 6.1 and the mini-mality of the cuts in question, one of the following possibilities holds: eitherC 0 = C(N0; "0) and C 00 = C(Nr; "00), or C 0 = C(N1; "0) and C 00 = C(Nr�1; "00).In the �rst case, the tight cut C(N1; "0) contains U inside (by Lemma 6.1),and thus both tight cuts in direction "0 contain U inside, a contradiction. Inthe second case, let us assume that there exists an edge " 2 P ("0; "00) distinctfrom "0 and "00. Then, by Lemma 6.1 and the minimality of C 0 and C 00, bothtight cuts in direction " do not contain a terminal U inside, a contradiction.Therefore, C0 and C00 are tight cuts of the same node N (N = N1 = Nr�1).One can prove easily that any other cut in I is also an N -tight cut.Moreover, any N -tight cut belongs to I. Indeed, assume that C is theN -tight cut in direction " and C =2 I (see Fig. 6.2b). Denote by C� the othertight cut in direction ". Then U lies either inside C, or inside C�. In the �rstTec

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43case, there exists a cut in I dominated by C. By the above reasoning, thiscut is an N -tight cut, and we thus get two N -tight cuts with nested innersides, which contradicts Corollary 6.2. In the second case, by Lemma 6.1,C� is dominated by all the N -tight cuts except for C, and thus by all thecuts in I, which contradicts the minimality of these cuts. Therefore, we canset �S(U) = N . �Thus, the projection mapping �S is de�ned, assigning to each unit Uthe path of Theorem 6.5 (observe that a single node is itself a path); theendpoints of this path are called the coordinates of U in HS . For an examplesee Fig. 5.1b,c: the coordinates of the unit U1 are N and N3, those of U2are N1 and N3, and those of U are N and N .The triple (FS;HS ; �S) is called the connectivity carcass of S. By Theo-rems 5.5(iii) and 5.6 its size is O(minf ~m;�S~ng) = O(minfm;�Sng).Let us consider a cut of the skeleton HS de�ned by a structural edge andrepresenting a bunch B. If we delete this edge from the skeleton, it fallsinto two connected components HS(B) and �HS(B) containing 'S(SB) and'S( �SB), respectively.Theorem 6.7. The set of vertices lying inside the tight (resp. outside theloose) cut of B is the union of all units U of FS such that �S(U) � HS(B)(resp. �S(U) � �HS(B)).Proof. Let us denote by " the structural edge corresponding to B, and byN and �N its endpoints lying in HS(B) and �HS(B), respectively. Then thetight cut of B is C(N; "), and the loose cut of B is �C( �N; "). Let now U bean arbitrary unit of FS . If " 2 �S(U), then, by de�nition, U lies outsideboth C(N; ") and C( �N; ").(ii) Otherwise, " =2 �S(U); by Theorem 6.5 we mayassume w.l.o.g. that �S(U) � HS(B). Let us prove that U lies inside C(N; ").Assume to the contrary that this is not the case; then U lies inside C( �N; "),since otherwise the projection of U would contain ". If U is a stretched unit,then there exists a structural edge "0 2 �S(U) � HS(B). Let ["0; "] =( �N 0; "0; N 0; : : : ; N; "; �N). Then U lies outside C(N 0; "0) and inside C( �N; "),in a contradiction to Lemma 6.1. Let now U be a terminal unit, and N 0 =�S(U) � HS(B) be its projection. We consider the link (N 0; "1; : : : ; N) ofN 0 and " in HS . By Theorem 6.5(ii), U lies inside C(N 0; "1), which, byLemma 6.1, contradicts to the fact that U lies outside C(N; "). �It follows readily from Theorem 6.7 that the projections of distinct ter-minal units are distinct (it su�ces to take for B the bunch of an arbitrarycut separating one of these units from the other). Hence, �S provides aninjection of the set of terminal units into the set of nodes of HS .Theorem 6.7 provides another point of view on the construction of The-orem 6.4.Corollary 6.8. A unit UN is the union of all units U of FS such that�S(U) � TN . A unit U of FS does not fall inside any unit UN , N 2Tec

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44L(S1; S2), if �S(U)\L(S1; S2) 6= ?. A unit UN is reachable from U if N isnot inner for �S(U) \ L(S1; S2).The following statement reveals an intimate relation between the projec-tion of a stretched unit and its reachability cones. Let U be a stretchedunit, R1 and R2 be the reachability cones of U , N1 and N2 be the sets ofnodes that are the projections of the terminals belonging to R1 and R2,respectively, T1 and T2 be the minimum subtrees of HS containing N1 andN2, respectively. Observe that each reachability cone of U contains at leastone terminal, and hence both T1 and T2 are nonempty.Theorem 6.9. The projection of U coincides with the link of T1 and T2.Proof. Let " be an arbitrary structural edge in the projection �S(U) andB be one of the two opposite bunches represented by ". It is easy to seethat all the terminals that belong to R1 lie inside the tight cut of B, whilethose in R2 lie outside the loose cut of B, up to ipping of B. Hence, byTheorem 6.7, T1 � HS(B) and T2 � �HS(B). It follows immediately that thetrees T1 and T2 are disjoint, and that �S(U) � L(T1; T2).Assume now that �S(U) 6= L(T1; T2); then there exists a subpath P =(N 0; "0; N; "00; N 00) of the link such that "0 belongs to �S(U), while "00 doesnot. By Corollary 6.8, UN is reachable from U in W(P ). Hence, by theremark after Theorem 6.4, there exists a coherent path in FS that leadsfrom U to the intersection of the inner sides of C(N; "0) and C(N; "00). Letus extend this path to a terminal U� of FS (see Fig. 6.3).U*

εε

)εC (N,

NN’ N"

C (N,ε )

εFig. 6.3. To the proof of Theorem 6.8If the extended path leaves the inner side of the N -tight cut in direction" for some " 6= "0; "00, then it cannot return back, and, by Theorem 6.7, theprojection of U� lies in the subtree of HS that is separated from N by ".However, this projection belongs either to T1, or to T2, and hence one of theedges "0 and "00 does not belong to L(T1; T2), a contradiction.Otherwise, U� lies inside all the N -tight cuts, and thus, by Theorem 6.5,it is projected to N , which gives the same contradiction as above. �By Theorem 6.9, each of T1 and T2 contains exactly one coordinate ofU . Therefore, we can label the reachability cones of any stretched unit (andTec

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45thus the parts of the inherent partition) by its coordinates; the reachabilitycone of U that contains the terminals whose projections coincide with or lie\behind" the coordinate N is denoted by RN(U). Moreover, we can nowredenote the trees Ti: the tree that contains N is denoted by TN(U).6.4. The cell structure. For any path on the skeleton with endpoints Mand N we de�ne the cell QMN as the set of units with the coordinatesM;N ;by de�nition, the cell QNM coincides with QMN . A cell of type QMM iscalled terminal ; it is either empty or consists of exactly one terminal unit(with projection M).Let us de�ne the �-reachability relation on cells. We say that a cellQM2N2 is �-reachable from a nonterminal cell QM1N1 in direction N1 if bothP1 = [M1; N1] and P2 = [M2; N2] are subpaths of the same path P so thatM1 coincides with one of the endpoints of P and at least one of M2, N2coincides with the other endpoint of P . We denote this path P by [P1; P2](which is consistent with the previous use of notation [�; �]). Evidently, ifQM2N2 is �-reachable from QM1N1 in both directions, then QM2N2 = QM1N1 .The following statement is a generalization of Lemma 4.11.Theorem 6.10. Let U1 be a stretched unit with coordinates M1 and N1,U2 2 RN1(U1), Q1 and Q2 be the cells containing U1 and U2, respectively.Then Q2 is �-reachable from Q1 in direction N1.Proof. If U2 is a terminal, then the assertion follows immediately fromTheorem 6.9. Let now U2 be a stretched unit with coordinates M2 andN2, and let w.l.o.g. U1 2 RM2(U2). Since RN2(U2) � RN1(U1), we getTN2(U2) � TN1(U1), hence the path [N1; N2] lies entirely in TN1(U1), and,by Theorem 6.9, intersects with [M1; N1] exactly by N1. Therefore, N1 lieson [M1; N2]. Since U2 is stretched, we can prove in a similar way that M2lies on [M1; N2], as required. �By Theorem 6.10, when we consider several units lying on the samecoherent path, we may assume that their coordinates are named consis-tently. In other words, if �S(U1) = [M1; N1], �S(U2) = [M2; N2], andU2 2 RN1(U1), then we assume that U1 belongs to RM2(U2), and hence[�S(U1); �S(U2)] = [M1; N2].Corollary 6.11. Let U lie on a coherent path P from U1 to U2, then M 2[M1;M2] and N 2 [N1; N2].Remark. It follows immediately from Corollary 6.11 that when one buildsthe strip W([M1; N2]) (as described in Theorem 6.4), no intermediate unitof P is subject to contraction.We now can extend the de�nition of the projection to the edges of FS ,similarly to what was done in Sect. 4.4. Let e = (U1; U2) be an edge of FS ,and assume that U1 2 QM1N1 , U2 2 QM2N2 . We the say that the projectionof e equals [M1; N2]; in other words, we de�ne �S(U1; U2) = [�S(U1); �S(U2)].Tec

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46As in Sect. 4.4, this de�nition can be justi�ed by inserting a dummy vertexin the edge e; it is easy to see that the projection of the unit that containssuch a vertex is exactly [M1; N2]. As before, there are three types of edgesbelonging to a cell QMN : internal (both endpoints belong to QMN ), external(exactly one endpoint belongs to QMN ), and through (both endpoints donot belong to QMN).For any proper path [M;N ], M 6= N , we build from FS a locally ori-entable graph WMN with the set of nonterminals QMN (which may be aswell empty). We delete all the cells non-�-reachable fromQMN (observe thatby Theorem 6.10 there are no edges between QMN and the deleted sets) andcontract all the cells �-reachable from QMN only in direction M and those�-reachable only in direction N into two terminals ~M and ~N , respectively.Finally, we delete all the edges between ~M and ~N that do not belong toQMN . An S-mincut C of G that separates the preimages of the terminalsand thus generates a ( ~M; ~N)-mincut ~C in WMN is called an extension of ~C.The following result is a generalization of Lemmas 4.2, 4.8 and 4.12.Theorem 6.12.(i) The graph WMN is a strip; moreover, if [M;N ] contains at least twostructural edges, then its width �MN does not exceed �S=2.(ii) All its transversal cuts, and only they, are extendable to S-mincuts.Such an extension can be found in any bunch corresponding to an edge in[M;N ] and in no other bunch.(iii) Let fQMiNig be a set of pairwise non-�-reachable cells such thatall the paths [Mi; Ni] contain at least one common edge. Then any set oftransversal cuts of WMiNi has a mutual extension to an S-mincut. More-over, if all the paths [Mi; Ni] contain at least two common edges, thenPi �MiNi 6 �S=2.(iv) If a 2-partition of V nSiQMiNi enters a mutual extension of somesample of transversal cuts, one from each of WMiNi , then it enters somemutual extension of any such sample. (Informally : samples of transversalcuts of these strips are interchangeable in S-mincuts).Proof. We start from the following general observation. Let "0 = (M 0; N 0) bean arbitrary structural edge of [M;N ] (we assume that [M;M 0] and [N;N 0]do not intersect). Let us consider the strip W("0). Its terminals correspondnaturally to M 0 and N 0. Evidently, any unit of QMN remains stretched inW("0); its cones and the sides of its star are labeled by M 0 and N 0. LetRM0 denote the union over U 2 QMN of the cones RM0(U) in W("0); RN0is de�ned similarly.Let us delete �RM0\ �RN0 and contract �RM0\RN 0 and RM0\ �RN 0 into twonew terminals ~M 0 and ~N 0, respectively. It is easy to see that each direct edgebetween ~M 0 and ~N 0 corresponds to a through edge in QMN ; the converseis not true, since the endpoints of a through edge do not necessarily belongto RM0 or RN 0 . Thus, the number of direct edges between ~M 0 and ~N 0 doesTec

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47not exceed the number of through edges in QMN . If it is strictly less, weadd complementary direct edges in order to equalize these quantities. Letus denote the obtained locally orientable graph by ~WMN . We claim that~WMN coincides with WMN .First of all, QMN = RM0 \ RN0 . Indeed, the inclusion QMN � RM 0 \RN0 is trivial. On the other hand, if U 2 RM 0 \ RN 0 , then there existU 0; U 00 2 QMN such that U lies on a coherent path from U 0 to U 00. Hence,by Corollary 6.11, �S(U) = [M;N ], and U 2 QMN .Let us consider an arbitrary edge e = (U; U�) such that U 2 QMN andU� =2 QMN . While constructing ~WMN and WMN , this edge is not deleted(by Lemma 2.1 and Theorem 6.10, respectively) and goes from U to a ter-minal. In ~WMN this terminal corresponds to the label of the side at U inFS that contains e. By Theorems 6.7 and 6.10, the same holds true forWMN . Therefore, the stars of the corresponding terminals in ~WMN andWMN coincide, and thus these locally orientable graphs coincide as well.(i) By the construction of ~WMN , it is a balanced acyclic two-terminallocally orientable graph, hence, by Strip Lemma(i), it is a strip.Let [M;N ] contain at least two structural edges, and let N1, N2, N3 beany three consequent nodes in [M;N ]. We consider the strip W([N1; N3])and de�ne the cones RN1 and RN3 in it and the strip ~WMN � WMN in thesame way as above. By Theorem 6.10 and the construction of W([N1; N3])in Theorem 6.4, the contracted unit UN2 belongs to �RN1 \ �RN3 . SinceC1 = (RN1 ; �RN1) and C3 = (RN3 ; �RN3) are S-mincuts, there are no edgesbetween RN1 \RN3 and �RN1 \ �RN3 (by Lemma 2.1). Next, since C1 \ �C3is an S-mincut, one has c(RN1 \ �RN3 ; �RN1 \ �RN3) = �S � �MN ; similarly,c( �RN1 \ RN3 ; �RN1 \ �RN3) = �S � �MN (see Fig. 6.4). Hence, c( �RN1 \�RN3 ;RN1 [ RN3) = 2(�S � �MN). However, since UN2 \ S 6= ?, one hasc( �RN1 \ �RN3 ;RN1 [RN3) > �S . Thus �MN < �S=2 as required.λMN

SλλMN

MNQ

NU NUNU 31 2

NR 1 NR 3

Fig. 6.4. To the proof of inequality �MN < �S=2(ii) Existence of an extension for any (M 0; N 0) 2 [M;N ] is proved in thesame way as it was done in Lemma 4.12(ii), with the set Xa replaced byRM0 \ �RN 0 and �Xc by RN0 \ �RM0 .Any extension of a transversal cut of WMN separates the preimages of~M from those of ~N , and hence, the set of terminals \behind" M from theTec

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48set of terminals \behind" N . By Fact 6.3(ii) and the second remark afterTheorem 6.4, the bunch of such a cut must be de�ned by an edge in [M;N ].(iii) Let "0 = (M 0; N 0) be a common edge of all the [Mi; Ni]. We considerthe strip W("0) and build the sets RiM0 and RiN 0 as above. Let R�M0 be theunion of RiM0 over all i, and let R�M0 be de�ned similarly. Let e = (U; U�)be an edge such that U 2 QMiNi , U� =2 QMiNi , and U� =2 RiM0 , thenU� =2 R�M0 . Indeed, assume to the contrary that U� 2 RjM0 , j 6= i, then, byTheorem 6.10, the cell QMiNi is �-reachable from QMjNj , a contradiction.The rest of the proof of the existence of a mutual extension is similar tothat of Lemma 4.12(iii), as above. The inequality for Pi �MiNi followsimmediately from Lemma 4.12(iii) applied to the strip W(\i[Mi; Ni]). Oneshould note, however, that some nodes of (\i[Mi; Ni]) can be empty, andhence the inequality of type �ii > �S=2 for such a node is replaced by�ii > �S=2.As a consequence, we get a nonstrict inequality for Pi �MiNi .(iv) The proof is similar to that of Lemma 4.12(iv). �6.5. Other properties of the connectivity carcass. Assume that weare interested in pointing out an S-mincut separating two given vertices ofG that lie in distinct units U1 and U2. The following statement is obtainedeasily.Lemma 6.13.(i) Let �S(U1) and �S(U2) be distinct nodes of HS . Consider any cut ofHS separating them, and let B be any of the corresponding bunches. Thenthe tight and the loose cuts of B separate U1 from U2.(ii) Let one of the projections, say, �S(U1), contain a structural edge"1 = (M1; N1), and let M , N be the coordinates of U1 in the correspondingorder. Then one of the sets RM(U1) [ V C(M1;"1) and RN(U1) [ V C(N1;"1)de�nes an S-mincut separating U1 from U2.Proof. (i) Follows from Fact 6.3(ii).(ii) It is clear from the structure of W"1 that both sets mentioned de�neS-mincuts; on the other hand, their intersection is exactly U1 6= U2, and theresult follows. �Consider now a pair s, t of non (�S + 1)-connected vertices in S. Let usdecompose the connectivity carcass with respect to this pair.Let Ui = UNi , 0 6 i 6 k, be the heavy units of Ws;t, s 2 U0, t 2 Uk (seeTheorem 6.4 for details); it is easy to prove that Ui are numbered accordingto any topological order of Ws;t. For any i as above we put �i = S \ Uiand contract the set S in two di�erent ways: Si is obtained by contractionof S n �i into a single vertex si, and ~S is obtained by contraction of eachof �i into a single vertex ~si. The graphs Gi and ~G are de�ned similarlyas the results of the above two contractions applied to G. Note that theconnectivity of ~S in ~G is exactly �S , while the connectivity of Si in Gi canbe greater than �S.Tec

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49It can be shown easily that the connectivity carcass of ~S in ~G has thesame type as one considered in Sect. 4.4. Its skeleton is the path [N0; Nk]and its esh is obtained fromWs;t by turning the heavy units Ui, 0 6 i 6 k,into terminals.Let I denote the set of indices such that �Si(Gi) = �S .Lemma 6.14.(i) The skeleton HS is obtained from the path [N0; Nk] = H ~S( ~G) and theskeletons HSi(Gi), i 2 I , by identifying Ni with the node N i = 'Si(si) ofHSi(Gi).(ii) The units of the esh FS are:- the units of FSi(Gi), i 2 I , except for the unit U i containing si;- the nonheavy units of Ws;t;- the terminal units U (i) = U i \ CNi;(Ni;Ni�1) \ CNi;(Ni;Ni+1) 6= ?, i 2 I ,and U (i) = CNi;(Ni;Ni�1) \CNi;(Ni;Ni+1) 6= ?, i =2 I , (for i = 0 and i = k onlyone of the above two cuts is taken).Moreover, if a unit other than U (i) participates in a more than one eshas above, then it participates in F ~S( ~G) and at most two of FSi(Gi).(iii) The projection of any unit other than U (i) is the concatenation of itsprojections in the corresponding connectivity carcasses.Proof. Observe that each S-mincut of G either partitions one of �i, i 2 I ,and does not partition its complement in S, or separates some of �i from theothers according to the skeleton HS . More exactly, each bunch of S-mincutsof G coincides either with a bunch of ~S-mincuts of ~G, or with a bunch ofSi-mincuts of Gi for some i 2 I . This implies assertion (i).Let U be a terminal unit of FS corresponding to a node N of HS . Recallthat U is the intersection of all N -tight cuts. If N belongs to the subtree Hiof HS hanging on the path [N0; Nk] at Ni and N 6= Ni, then all these cutsare N -tight cuts of the connectivity carcass of Si in Gi. Therefore, U is aterminal unit of this partial connectivity carcass. If N = Ni, i 2 I , then allN -tight cuts are of the same type, except for one or two cuts correspondingto the edges of [N0; Nk] incident to Ni. Clearly, Ui is a unit of FS for anyi 2 I . Thus we get assertion (ii) for terminals.Let U be a stretched unit. It participates in all bunches that correspondto the edges of its projection. The projection is contained either in one ofHi, or in Hi [ [N0; Nk] for some i, or in Hi [ [Ni; Nj] [ Hj . The rest ofassertion (ii) follows, as well as assertion (iii). �7. Incremental transformations of the connectivity carcass7.1. Transformations of the components of the connectivity car-cass. In this section we describe the transformations of all the three compo-nents of the connectivity carcass caused by insertion of a new edge (u1; u2)that preserves the value of �S . In what follows Ui stands for the unit con-taining ui, Qi for the cell containing Ui, and Pi for the projection �S(Ui),Tec

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50i = 1; 2. If U1 = U2, then the carcass does not change. So in what follows weassume that U1 6= U2. If P1 and P2 are edge-disjoint, we denote by L1 andL2, respectively, the endpoints of the link L = L(P1; P2). Otherwise, we setL = ? and denote by [M;N ] the intersection of the projections; furthermore,in this case we assume w.l.o.g. that if P1 = [M1; N1] and P2 = [M2; N2],then the paths [M1;M2] and [N1; N2] do not intersect and M 2 [M1;M2],N 2 [N1; N2]. In both cases we denote by T the minimal subtree of HScontaining both P1 and P2. Recall that we put hat over any notation (e.g.,bHS , or bFS) to denote the corresponding object after edge insertion.The main concern of this Section is to distinguish, in terms of the connec-tivity carcass, the S-mincuts that do not separate U1 from U2, since theseare exactly the S-mincuts that are preserved under the insertion.The transformations of the skeleton under edge insertion are as follows.Theorem 7.1. If L contains at least one structural edge, then all the nodesand structural edges of L are contracted to a single new node; otherwise theskeleton remains the same.Proof. Let bHS be obtained from HS by contraction of L into a new nodeN (or bHS = HS if L does not contain any edges). We de�ne b'S as follows:b'S = 'S if 'S =2 L and b'S = N otherwise.Let " be an arbitrary structural edge in HS .If " =2 T , then, by Theorem 6.7, both U1 and U2 lie in the same terminalof the strip W("). Hence, in this case the new edge does not a�ect any cutof this strip.If " 2 T , but " =2 L, then at least one of the units U1 and U2 is astretched unit in W("). Hence, in this case at least one tight cut of W(") isnot a�ected by the new edge, and thus the corresponding 2-partition of S isstill an S-mincut.Finally, if " 2 L, then, by Theorem 6.7, U1 and U2 lie in distinct terminalsof W("). Hence, in this case no cuts represented by W(") survive, and thusthe corresponding 2-partition of S ceases to be an S-mincut.Thus, by Theorem 5.6, bHS is indeed the skeleton of the graph G [(u1; u2). �The transformations of the esh under edge insertion are as follows.Theorem 7.2. The new esh is obtained by contraction of a certain subsetof units of FS into a single new unit Unew. This subset contains :(i) U1 and U2;(ii) all the units lying on coherent paths between U1 and U2;(iii) all the units U such that �S(U) � L;(iv) in the case when an endpoint K of Pi is the unique common point ofPi and Pj [L, i; j 2 f1; 2g, i 6= j, all the units U such that U 2 RK(Ui) and�S(U) � Pi [ L.Tec

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51Proof. First, let us prove the following statement concerning the initial graphG.Claim A. If a unit U satis�es at least one of the conditions of Theorem 7.2,then any S-mincut that separates U1 from U separates also U1 from U2.Indeed, if U = U2, then the claim is trivial.Let U belong to a coherent path between U1 and U2. Let C be an S-mincut that separates U1 from U but fails to separate U1 from U2, then weget a coherent path with both endpoints on one side of C and an intermediateunit U on the other side, a contradiction.Let �S(U) � L, and let C be an S-mincut separating U1 from U . ByTheorem 6.7, the structural edge " representing C in HS lies either in L, orin P1. If " 2 L, then U1 and U2 are separated by any cut represented by" (see the proof of Theorem 7.1). If " 2 P1, then U1 is a stretched unit ofW("), while both U and U2 lie in the same terminal of W("), and the claimfollows.Assume now that an endpoint K of P1 is the unique common point of P1and P2 [ L, and besides, U 2 RK(U1) and �S(U) � P1 [ L. Let C and "be as in the previous case, then, by Theorem 6.7, " belongs to P1 [ L. If" 2 L, we proceed as above. If " 2 P1 \ �S(U), then both U1 and U arestretched units of W("). Moreover, U2 lies in one of the terminals of W("),and, by Theorem 6.10, U lies in the cone of U1 containing this terminal.Thus, C separates U1 from U2. If " lies in P1, but not in �S(U), then, byTheorem 6.10, both U and U2 lie in the same terminal of W("), and againC separates U1 from U2.Finally, if an endpoint K of P2 is the unique common point of P2 andP1[L, and besides, U 2 RK(U2) and �S(U) � P2[L, one has to interchangethe roles of U1 and U2 in the above reasoning.Therefore, Claim A is proved, and as an immediate corollary we get thatall the units listed in Theorem 7.2 are indeed contracted into a single newunit.Let us now prove another statement concerning the initial graph G.Claim B. If a unit U violates all the conditions of Theorem 7.2 and U 0 6= Uis an arbitrary unit, then there exists an S-mincut that separates U from U 0and does not separate U1 from U2.Indeed, let P = �S(U), Q be the cell containing U , and assume �rst thatU is a stretched unit.Suppose that there exists a structural edge " 2 P that does not belong toT . Then U is a stretched unit in W("), while both U1 and U2 lie in the sameterminal ofW(") (by Theorem 6.7). Therefore, any S-mincut that separatesU from U 0 in W(") satis�es the assertion of the claim.From now on we may assume that P lies entirely in T . By the assumptionsof the Theorem, P does not lie entirely in L, hence, there exists an edgeTec

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52" 2 P that belongs (w.l.o.g.) to P1. If " belongs also to P2, then all thethree units U , U1, and U2 are stretched in W("). If U1 and U2 lie in theopposite cones of U in W("), then U lies on a coherent path from U1 to U2,which is prohibited by the assumptions of the claim. So, there exists a coneR of U in W(") that does not contain neither U1, nor U2. If R does notcontain U 0 as well, then the cut of W(") de�ned by R satis�es the assertionof the claim. Otherwise, we use instead of R the cone R0 of U 0 in W(") thatis contained in R.If " =2 P2, then in W(") both U and U1 are stretched, while U2 lies in aterminal. Let R be the cone of U in W(") that contains the other terminal.If U1 does not belong to R, then the cut of W(") de�ned by R (or by R0 asabove, if U 0 2 R) satis�es the assertion of the claim.Let now U1 2 R; this means, in particular, that U belongs to some coneof U1 in W("), and thus in FS ; say, U 2 RK(U1). By Theorem 6.10, thismeans that Q is �-reachable from Q1 in direction K. Therefore, if "1 is theedge of P1 incident to K, then "1 2 P . If "1 2 P2, we proceed in the sameway as above; so, we may assume that "1 =2 P2. Let us consider the stripW("1). Both U and U1 are stretched in W("1), moreover, U lies in a coneR1 of U1, while U2 lies in a terminal. If this terminal does not belong toR1, we consider the cone R of U that lies inside R1. It is easy to see thatthe cut of W("1) de�ned by R (or by R0 as above, if U 0 2 R) satis�es theassertion of the claim.Finally, let the terminal containing U2 in W(") lie in R1. It followsimmediately from Theorem 6.7 that in this case K is the unique commonpoint of P1 and P2 [ L. By the assumptions of the claim, P does not lieentirely in P1 [ L, hence, there exists and edge "2 of P such that "2 2 P2.Evidently, inW("2) both U and U2 are stretched, while U1 lies in a terminal.Let U does not lie in the cone of U2 in W("2) that contains this terminal,and let R be the cone of U in W("2) that contains the other terminal; thenthe cut of W("2) de�ned by R (or by R0 as above, if U 0 2 R) satis�es theassertion of the claim. Otherwise, we use the same reasoning as above tosee that one of the endpoints of P2 is the unique common point of P2 andP1 [ L, and hence T is just a path. Therefore, Q1 and Q2 are �-reachablefrom Q in opposite directions. Thus, the coherent path from U1 to U inW("1) can be concatenated with a coherent path from U to U2 in W("2),and we get that U lies on a coherent path from U1 to U2, in a contradictionwith the assumptions of the claim.Assume now that U is a terminal unit, and let P 0 = �S(U 0). Observethat P from now on is a node of HS .If P \ P 0 = ?, then, by the assumptions of the claim, there exists astructural edge " that belongs to the link of P and P 0 and does not lie in L.Evidently, the units U and U 0 lie in the opposite terminals of the stripW(").On the other hand, the units U1 and U2 cannot lie in the opposite terminalsin such a strip. Therefore, one of the tight cuts in direction " satis�es theTec

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53assertion of the claim.Let now U 0 = U1 and P be a node of P1. If P is not an endpoint ofP1, then at least one of the two P -tight cuts in directions of the edges ofP1 incident to P does not contain U2 (otherwise P would belong to L). IfP is an endpoint of P1, then the P -tight cut in direction of the edge of P1incident to P does not contain U2 (for the same reason as above).This completes the proof of Claim B, since if U 0 is stretched and satis�esthe assumptions of the claim, then we can use the same reasoning as aboveinterchanging the roles of U and U 0, whereas if U 0 violates the assumptionsof the claim, then any cut that separates U from U1 separates also U fromU 0, by Claim A.As an immediate corollary of Claim B we get that all the units not listedin Theorem 7.2 remain uncontracted, and thus the Theorem is proved. �The transformations of the projections under edge insertion are as follows.Theorem 7.3.(i) The projection of Unew is either the new node obtained by the contrac-tion of L (if L contains at least one structural edge), or P1 \ P2, otherwise.(ii) The intersection with L is deleted from the projection of any noncon-tracted unit.(iii) For each unit U 2 RN1(U1), the common part of �S(U) and the pathbetween M1 and M (or L1) is deleted from �S(U). Similar transformationsare applied to RM1(U1), RN2(U2), RM2(U2).Proof. Assume �rst that U is an arbitrary stretched unit of the initial graph,and let " be an arbitrary structural edge of �S(U).If " =2 T , then both U1 and U2 lie in the same terminal ofW("), and thusthe new edge does not a�ect any cut of W("). Hence, " is preserved in thenew projection of U .If " 2 L, then U1 and U2 lie in the opposite terminals ofW("), and thus allthe cuts ofW(") do not survive. Hence, " is deleted from the new projectionof U .If " 2 P1\P2 , then both U1 and U2 are stretched units ofW("). Therefore,both tight cuts of W(") remain una�ected by the new edge, and hence " ispreserved in the new projection of U .The only remaining case is when " belongs to exactly one of the P1 andP2; assume w.l.o.g. that " 2 [M1; K], where K = M if P1 \ P2 6= ?, andK = L1 otherwise. Then U1 is a stretched unit of W("), while U2 lies inthe terminal ofW(") that belongs to RN1(U1) (or, more exactly, to the coneof U1 in W(") that lies in RN1(U1)). Therefore, if U =2 RN1(U1), then " ispreserved in the new projection of U , while otherwise it is deleted from thenew projection.Assume now that U is a terminal of FS ; evidently, the new edge cannotconvert it to a stretched unit, and thus all we have to do is to establish thatthe new projection of U cannot shift.Tec

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54By Theorem 6.5(ii), U lies in the intersection of all the K-tight cuts,where K = �S(U). It follows easily from Theorem 7.1 that if " is incidentto K and " =2 L, then the new K-tight cut in direction " dominates the oldone. Therefore, if K =2 L, or K = L, then K remains to be the projectionof U in the new skeleton.Finally, let K 2 L and K 6= L; we denote by "i and "0i the edges of thepath [K;Li] incident to K and Li, respectively (i = 1; 2). By Lemma 6.1,any Li-tight cut, except for the Li-tight cut in direction "0i, dominates theK-tight cut in direction "i. By Theorem 7.1, all the edges incident to Liexcept for "0i are preserved in the new skeleton, and are incident to its newnode. Evidently, the tight cut of the new node in any of these directionsdominates the Li-tight cut in the same direction. Thus, the Unew lies in theintersection of all the tight cuts of the new node. �Remark. It follows from Theorems 7.2 and 7.3 that if a unit must be con-tracted, then its new projection obtained formally via Theorem 7.3 coincideswith that of Unew.Finally, let us describe the transformation of the inherent 2-partition un-der edge insertion.Theorem 7.4.(i) The inherent 2-partitions at noncontracted units remain the same.(ii) If P1 \P2 does not contain any edges, then the inherent 2-partition atUnew is trivial. Otherwise, the inherent 2-partition for Unew is glued fromthose for contracted units. Namely, the part of the star of U1 labeled by N1(resp. M1) is glued with that for U2 labeled by N2 (resp. M2). For otherunits, two parts are glued together if the corresponding reachability conescontain the same unit out of U1 and U2.Proof. (i) Follows immediately from Theorem 7.3, since the 2-partition at aunit U of the esh is either trivial, or inherited from the 2-partition at thesame unit of any strip cW("), " 2 �S(U).(ii) The �rst part follows immediately from Theorem 7.3.Let now P1 \ P2 contain at least one edge ". By Lemma 5.3, it is enoughto prove the statement for the 2-partition at U in the strip cW("). It followsfrom Theorems 7.1-7.3 that to get cW(") fromW(") one has just to insert theedge (U1; U2) intoW(") and to transform it accordingly. This transformationwas already considered in Sect. 4.1, so the statement follows. �7.2. Local reachability cones. The transformations of the connectivitycarcass described in Theorems 7.2 and 7.3 are formulated in terms of reach-ability cones. Therefore, to maintain the carcass, we need to maintain thecones as well. However, reachability cones in an acyclic locally orientablegraph behave in a more complicated way than those in a dag. In particular,insertion of an edge may cause a reduction of a reachability cone. Indeed,if a stretched unit belonging to a cone is contracted into a terminal (seeTec

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55Theorem 7.2), then any coherent path passing through this unit cannot betraced behind it in a modi�ed esh. To avoid these di�culties, we introducea simpler structure, called a local reachability cone, which preserves all theuseful properties of reachability cones, but is easier to handle.Let R be a reachability cone of an arbitrary unit U . We de�ne the localreachability cone Rloc as the set of units U 0 2 R such that �S(U) \ �S(U 0)contains at least one edge (in particular, the local cone of a terminal isempty). One can check easily that local reachability cones of U are obtainedfrom the corresponding reachability cones of U in the strip W(�S(U)) bydeleting the terminals. Observe that reachability cones in the assumptions ofTheorems 7.2 and 7.3 can be replaced by the corresponding local reachabilitycones.Indeed, for any unit U whose projection is changed according to The-orem 7.3(iii), the intersection �S(U) \ P1 contains at least one edge, andhence such a unit belongs to RlocN1(U1) (or, similarly, to RlocM1(U1), RlocN2(U2),or RlocM2(U2)).The units U distinguished by condition (iv) of Theorem 7.2 satisfy U 2RK(Ui) and �S(U) � Pi [ L. However, if �S(U) \ Pi does not containany edges, the latter inclusion implies �S(U) � L, and hence U is alreadydistinguished by condition (iii). Otherwise �S(U) \ Pi contains at least oneedge, and this together with the former inclusion implies U 2 RlocK (Ui).Finally, let us consider the units distinguished by condition (ii) of Theo-rem 7.2. Evidently, these units form the intersection of the two reachabilitycones R(U1) and R(U2) (for the sake of simplicity, we do not specify thelabels of the cones). Let us prove that it su�ces to consider only the unitslying in the intersection Rloc(U1)\Rloc(U2). Indeed, if R(U1)\R(U2) = ?,then Rloc(U1) \ Rloc(U2) = ?, and the assertion is trivial. Otherwise, byTheorem 6.10, the minimum tree containing P1 and P2 is a path, and theprojection of any unit U 2 R(U1) \ R(U2) belongs to this path. If �S(U)intersects both P1 and P2 at least by an edge, then U 2 Rloc(U1)\Rloc(U2).If �S(U) intersects both P1 and P2 at most by a node, then �S(U) � L,and U is distinguished by condition (iii) of Theorem 7.2. Finally, if �S(U)intersects at least by an edge exactly one of P1 and P2, then the assumptionsof Theorem 7.2(iv) are satis�ed, and U is distinguished by condition (iv).8. Construction and incrementalmaintenance of the connectivity carcass8.1. Construction of the connectivity carcass. We build the connec-tivity carcass by a recursive algorithm based straightforwardly on Lemma 6.14;in what follows we use the notation of this lemma.First of all, we choose an arbitrary vertex s 2 S and �nd maximal owsfrom s to all the other vertices t 2 S; this allows to de�ne �S as the minimalvalue of these ows. All the vertices t 2 S such that the value of a maximal ow from s to t exceeds �S are contracted together with s; in what followsTec

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56s denotes this new vertex. We then choose an arbitrary of the remainingvertices of S (denoted by t) and execute the following procedure.We build the strip F = Ws;t as explained in Sect. 3.2. To build theskeleton H = H ~S( ~G) we �nd a topological order of the units in F . LetU0 3 s; U1; : : : ; Uk 3 t be the heavy units in this order; we de�ne H as ak-edge path (N0; N1; : : : ; Nk), k > 1.To �nd the corresponding projections � = � ~S , we execute DFS in F indirection Uk �rst from Uk�1, then from Uk�2, and so on up to U0. In anysuch execution we assign Ni as a coordinate to the units found from Ui andbacktrack each time when we discover a unit already visited in the previoussearch. To get the other coordinate we repeat the same process in directionU0 �rst from U1, then from U2, and so on up to Uk.Next, for each i, 0 6 i 6 k, we turn the unit Ui into a terminal. If Uicontains vertices of S distinct from s and t, we do the following. We buildthe set S0 by taking all the vertices of S that do not belong to Ui, togetherwith s and t, if they are not yet included in S 0. We contract all the verticesof S0 into a new vertex si, thus obtaining the new graph Gi and the new setSi in it. We then scan the vertices t0 in Si n si and �nd a maximum owfrom si to t0. If its value exceeds �S , then we contract t0 to si. Otherwise,we stop scanning and build the connectivity carcass (Hi;F i; �i) of Si in Giby a recursive execution of the same procedure, with s = si, t = t0 . Thiscarcass is then merged with the current triple (H;F ; �) as follows.The skeleton H is merged with Hi by identifying the node Ni 2 H withthe node �i(si) 2 Hi.The new terminal unit corresponding to Ni is built according to Lem-ma 6.14(ii).To merge the partitions of V into units of F and F i we assume thateach vertex v 2 V knows its units U(v) and U i(v), and each unit knows itscardinality. Now, for each v 2 V we check the units U(v) and U i(v). Iftheir cardinalities are equal, we concatenate their projections into the newprojection of U(v) and cancel U i(v). Otherwise, we set the new U(v) to bethe smallest of the two units involved, preserve its projection, and cancelthe other unit.The above described algorithm implies the following result (the proof isomitted).Theorem 8.1. The connectivity carcass of an arbitrary vertex subset Scan be constructed in time dominated by the complexity of 2� � 2 max- owcomputations in G.8.2 Incremental maintenance of the connectivity carcass. Let n bethe number of vertices in the initial graph G, ~� 6 � be the number of(�S+1)-connectivity classes in S, ~n 6 n be the initial number of esh units,~m 6 m, be the initial number of esh edges.Tec

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57We maintain the connectivity carcass under an arbitrary sequence of up-dates (edge insertions) not changing the connectivity of S and queries \Arevertices u; v 2 G separated by an S-mincut?", denoted by Sep(u; v), \Showan S-mincut separating u; v 2 G", denoted by Cut(u; v), and \Constructthe strip WS1;S2 for S1; S2 � S", denoted by Strip(u; v).In order to maintain the connectivity carcass e�ciently, we propose tokeep track of the projections (in fact, the coordinates) and of certain parts ofreachability cones speci�ed below. It follows from the discussion in Sect. 7.2that instead of a reachability cone R it su�ces to maintain any partialsubcone Rpart of it that contains the corresponding local reachability coneRloc. By Lemma 3.4, a reachability cone R of an arbitrary unit W , andhence Rpart, is globally orientable. Since it is acyclic, it can be treated inthe same way as a dag. In what follows we assume that W is the source inthis dag. Following [I], we represent Rpart by a directed spanning tree rootedat W and make use of the reachability vector of length ~n. Each entry of thevector takes one of the three values to distinguish the following situations:the unit is currently contained in Rpart; the unit was never contained inRpart; the unit was previously deleted from Rpart. At the initial moment weset Rpart = R, which takes O(~n ~m) time for all the cones together.We represent the skeleton as a rooted tree with an arbitrary �xed rootand make use of the data structure proposed in [W92]. This data structureallows to perform an arbitrary sequence of w Nearest Common Ancestor(NCA) queries and edge contractions in O(w+ t log2 t) time, where t is thesize of the initial tree. In our case this means that we can perform NCAqueries in O(1) amortized time with an overhead of O(~� log2 ~�), since byTheorem 5.6 the size of the skeleton is O(~�).The set of units is represented with the help of the standard union-�ndtechnique (see [AHU]) that allows to perform an arbitrary sequence of woperations in O(w+ t log t), where t is the size of the initial set. In our casethis means that we can perform �nd operations in O(1) worst-case time withan overhead of O(~n log ~n). Each side of a unit W is represented as a linkedlist of edges, and the whole list is labeled by the corresponding coordinateof W .When a new edge is inserted, it takes O(1) worst-case time to locate theunits U1 and U2 containing its endpoints. The nontrivial case, when U1 andU2 are distinct, can occur at most ~n times (since this causes the contractionof these units). For each nontrivial case we do the following.On the �rst stage we analyze the relative position of the projections�S(U1) and �S(U2). If they are edge-disjoint, we �nd their link and, ac-cording to Theorem 7.1, contract the edges of the link. Otherwise, we �ndtheir intersection and establish the correspondence between the endpointsof the intersection and the endpoints of the projections (see the beginningof Sect. 7.1 for details). It is easy to check that in order to �nd the linkor the intersection of two projections, and to establish the latter correspon-Tec

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58dence, it su�ces to apply a constant number of NCA queries. Since wespend O(1) amortized time for each of O(~n) NCA queries with an overheadof O(~� log2 ~�) (covering all contractions), the total time for the �rst stage isO(~� + ~n+ ~� log2 ~�) = O(~n+ ~� log2 ~�).On the second stage we change the projections. Evidently, the total costof projection changes prescribed by Theorem 7.3(i) is O(~n). Further, sinceprojections are represented by coordinates, contractions in the skeleton donot imply explicit changes in this representation. Thus, projection changesprescribed by Theorem 7.3(ii) does not require any time.Finally, to �nd units whose coordinates must be changed according toTheorem 7.3(iii), we execute DFS in the spanning tree of each of the fourpartial cones involved and backtrack each time when we reach a unit whoseprojection no more intersects a certain path in the skeleton, see Theo-rem 7.3(iii) for details (e.g., for the case of RN1(U1) this path is either[M1;M ] or [M1; L1]). The validity of such a backtracking is justi�ed byTheorem 6.10. It is easy to see that all the operations performed on thesecond stage, other than the analysis of the relative position of two pathsin the skeleton, require O(1) amortized time per unit. The latter analysiscan be performed in O(1) amortized time per unit as well by means of NCAqueries as above.To estimate the total amount of time required by projection changes inthis case, let us assign a weight to each edge in the esh. The weight of anedge is equal to the sum of the lengths of the projections of its endpoints(the length of a projection is just the number of structural edges in it).Observe that each time when an edge is scanned, the projection of its tail istruncated. Hence, the length of the projection of the tail strictly decreases,and the same occurs to the weight of the edge considered. Since the initialweight of each edge is O(~�), we see that the overall number of projectionchanges is O(~�~n), and that of edge scans is O(~� ~m). Thus, the total amountof work on the second stage is O(~n+ ~�~n+ ~� ~m+ ~� log2 ~�) = O(~� ~m).The contractions in the esh are performed on the third stage. Evidently,the total cost of contractions described in Theorem 7.2(i) is O(~n log ~n). Theset of units that must be contracted according to Theorem 7.2(ii) is theintersection of two opposite local reachability cones (see the discussion inSect. 7.2). To �nd these units we scan the reachability vectors of the cor-responding partial cones, and for each unit that belongs to the intersectionof the cones, compare its projection with �S(U1) and �S(U2). Since eachcontraction itself takes O(log ~n) amortized time, the total amount of timefor contractions in this case is O(~n2 + ~n log ~n+ ~� log2 ~�) = O(~n2).To �nd the units that must be contracted according to Theorem 7.2(iii)and (iv), we scan all the units and check whether their new coordinatescoincide with that of the contracted unit (see the remark after Theorem 7.3).This takes O(1) amortized time per unit (via techniques of [W92]), andthus O(~n) time for the whole scan. Each contraction itself takes O(log ~n)Tec

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59amortized time. Since each occurrence of this case leads to a contraction inthe skeleton, the total number of such occurrencies is O(~�). Hence, the totalamount of work in this case is O(~�~n+ ~n log ~n+ ~� log2 ~�) = O(~�~n+ ~n log ~n).To �nd the 2-partition at the new unit, provided it is stretched, we es-tablish the correspondence between the labels of the sides involved and con-catenate the corresponding linked lists. This can be done in O(~n+ ~� log2 ~�)time with the same technique as above.On the fourth stage we change partial reachability cones. Let Rpart bea partial cone of some unit W . As it was described above, Rpart is rep-resented by a spanning tree rooted at W and the three-valued reachabilityvector. Since the set of the contracted units is already known, we scan thecorresponding entries of the reachability vector and �nd the list L(Rpart) ofthe units in Rpart that should be contracted. Since each unit is contractedat most once, the total amount of work for these scans is O(~n2). If the listL(Rpart) is empty, the cone Rpart is not changed. Otherwise we proceed asfollows (see the proof of Theorem 8.2 for support).If Unew is a terminal, we just delete all the units belonging to L(Rpart),together with their subtrees, from the tree representing Rpart, and changethe reachability vector accordingly. Since each unit is deleted at most oncefrom each tree, and the total number of trees is O(~n), the total amount ofwork in this case is O(~n2).If Unew is not a terminal, we do the following. For each unit U belongingto the list L(Rpart) and distinct fromW we check whether its parent belongsto the list; if it does, we contract the edge from U to its parent, and if not,we mark this edge. If there are marked edges, we identify all their tailsinto a new unit W 0 and delete all marked edges except for an arbitraryone. Finally, if exactly one of U1 and U2 (say, U1) never belonged to Rpart,the other one (in our case, U2) belongs to Rpart, and �S(W ) \ (�S(U1) \�S(U2)) contains at least one edge, we add a certain subtree to Rpart byidentifying its root with W 0, if de�ned, or with W otherwise. To obtain thissubtree we execute DFS in the spanning tree of the corresponding partialcone of U1 and backtrack each time when we get to a unit that belongs toRpart or was previously deleted from Rpart. Clearly, the total amount ofwork in this case is proportional to the number of edge operations, that is,contractions, marking and deletions of edges currently in Rpart and scans ofedges currently not in Rpart. Each edge in Rpart is contracted or marked anddeleted at most once. Besides, each time when Rpart is updated, at most oneedge is marked and not deleted. Finally, each scan turns an edge currentlynot in the entire cone R to an edge in R. Moreover, such an edge neverbelonged to R before, hence each edge is scanned at most once. Therefore,the number of edge operations is O(~n ~m).To �nd whether two vertices of G are separated by an S-mincut, it suf-�ces to �nd the corresponding units (in O(1) worst-case time) and to checkwhether they do not coincide. If this is the case, the partition of S corre-Tec

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60sponding to such a cut can be obtained via Lemma 6.13 using Theorem 6.7.Finding a cut of the skeleton separating two given nodes and �nding an edgeof a projection takes O(1) amortized time (with the help of NCA queries).To �nd a tight cut as in Lemma 6.13(ii), we check for all the units theinclusion of Theorem 6.7 in O(1) amortized time per unit.To obtain the stripWS1;S2 of Theorem 6.4 we execute O(jS1j+ jS2j) NCAqueries to check whether T (S1) \ T (S2) = ?, and if this is the case, to �ndthe link L(S1; S2). Next, for each unit U we check whether its projectionintersects the path L(S1; S2) by an edge. If this is not the case, a few NCAqueries give us the endpoint N 2 L(S1; S2) of the link of L(S1; S2) and�S(U). The unit U belongs to the contracted subset corresponding to N .To obtain the orientation, it su�ces to execute DFS by coherent paths fromany terminal unit of the strip.The complexity of the above described algorithm is given by the followingstatement.Theorem 8.2. The connectivity carcass of an arbitrary vertex subset S canbe maintained in O(~n ~m+u+qsep+qcut~n+qstrip ~m) time for an arbitrary se-quence of u edge insertions preserving the value of �S , qsep queries Sep(u; v),qcut queries Cut(u; v), and qstrip queries Strip(u; v). Moreover, each querySep(u; v) can be answered in O(1) worst-case time.Proof. It follows from the discussion above that the only thing one has tocheck is that partial reachability cones are maintained properly, that is, thatat any moment they remain intermediate between local reachability conesand ordinary ones. Since for a terminal both the local cone and its partialcone maintained by the algorithm are trivial, we assume from now on thatW is a stretched unit and remains stretched in the modi�ed esh.It is convenient to use the same notation as in Sect. 7, that is, P1 =[M1; N1] and P2 = [M2; N2] are the projections of U1 and U2, respectively,[M;N ] is their intersection (it contains at least one edge by our assumptionconcerning W ), the paths [M1;M2] and [N1; N2] are disjoint, and M 2[M1;M2], N 2 [N1; N2]. Besides, b�S(U) stands for the modi�ed projectionof U , provided U exists in the modi�ed esh, or for the projection of Unewotherwise. In the latter case U is one of the units constituting Unew, andhence is implicitly contained in it as a set of vertices, which justi�es ournotation. In the same sense, we say that a cone is increased upon edgeinsertion if the set of vertices that constitute its units increases.Let us prove �rst the following statement.Lemma 8.3. If an edge insertion takes a unit U out of a local cone Rloc ofW , then U will never belong to Rloc again.Proof. We prove, moreover, that if U 2 Rloc and U =2 bRloc, then alreadyb�S(W ) and b�S(U) intersect at most by a node; since the projections canonly shrink upon edge insertions, this would mean that U never belongs toTec

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61Rloc again. Assume to the contrary that b�S(W ) and b�S(U) intersect atleast by an edge. Then U =2 bRloc can occur only if Unew is a terminal andeach coherent path from W to U in FS passes through at least one unitthat constitutes this terminal. Let U 0 be such an intermediate unit on acoherent path from W to U , and let b�S(U 0) = N . It follows immediatelyfrom Theorem 6.10 that b�S(W ) and b�S(U) belong to distinct branches ofbHS hanging at N , so they intersect at most by a node, a contradiction. �Remark. Observe that the monotonicity property of Lemma 8.3 does nothold for the cone R itself.It follows immediately from Lemma 8.3 that when maintaining Rpart, onedoes not need to include in it anew the units that were previously deletedfrom it.Observe that exactly the same reasoning proves that if Unew is a terminal,then the whole subtree rooted at any unit in L(Rpart) does not belong tothe corresponding bRloc. Thus, the case when the new unit is a terminal iscompleted.Assume now thatUnew is stretched; clearly, in this caseR is not decreased.First of all, observe that if U 0; U 00 2 L(Rpart) and there exists a coherentpath from W to U 00 passing through U 0, then all the units on this path lyingbetween U 0 and U 00 belong to L(Rpart) as well. Indeed, let U be such a unit,and consider an arbitrary S-mincut separating U , say, from U 0. It followsimmediately from Theorem 5.5(ii) that the same S-mincut separates also U 0from U 00, and hence it no more exists after the edge insertion. Therefore,U is contracted into Unew as well. This means that the units in L(Rpart)form a set of subtrees in the tree representing Rpart in such a way thatno root of a subtree is a descendant of another root. It follows immediatelythat the contraction-marking-deletion-identifying procedure described in thealgorithm is well de�ned, and that the tree thus obtained represents the partof bRpart containing all the vertices constitutingRpart, and thusRloc. Clearly,if Rloc does not increase, we are done.Let us consider the case when Rloc is increased. Evidently, this happensif and only if the new unit belongs to bRloc and at least one of U1 and U2(say, U1) does not belong to Rloc. Let us prove the following statement.Lemma 8.4. The conditions Unew 2 bRloc and U1 =2 Rloc are equivalent tothe following three: U1 =2 Rpart, U2 2 Rpart, and �S(W )\ (P1 \P2) containsat least one edge.Proof. Indeed, Unew 2 bRloc implies that b�S(W )\ (P1\P2) contains at leastone edge, therefore, the same is true for �S(W )\(P1\P2) (since projectionscan only shrink). Next, if �S(W ) \ (P1 \ P2) contains at least one edgeand U1 =2 Rloc, then U1 =2 R; indeed, U1 2 R would imply together withU1 =2 Rloc that �S(W ) intersects P1 at most by a node, a contradiction.Clearly, U1 =2 R implies U1 =2 Rpart. The same reasoning with U1 replacedby U2 shows that if U2 =2 Rpart, then U2 =2 R. To get �nally the conditionTec

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62U2 2 Rpart, it remains to rule out the case U1; U2 =2 R. It follows easilyfrom Theorem 7.2 that the units constituting Unew are, apart from U1 andU2, exactly those lying on coherent paths between U1 and U2. Therefore,U1; U2 =2 R together with Unew 2 bRloc would imply the existence of a unitU 2 R lying on such a path; hence, extending a coherent path from W toU behind U we can get either to U1, or to U2, a contradiction.In the other direction, U1 =2 Rpart implies U1 =2 Rloc and U2 2 Rpartimplies Unew 2 bR. By Theorem 7.3, any edge contained in �S(W )\(P1\P2)is contained also in b�S(W ) \ (P1 \ P2), hence, the existence of an edge inthe former intersection together with Unew 2 bR implies Unew 2 bRloc. �In fact, in the algorithm, instead of checking condition U1 =2 Rpart, wecheck whether U1 has never belonged to Rpart. Indeed, if U1 currently doesnot belong to Rpart but has belonged to it previously, then by Lemma 8.3we do not need to include U1 into bRpart.W U

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partRFig. 8.1. To the dynamics of RpartLet us describe now the additional units acquired by bRloc. Assume w.l.o.g.that the cones of W under consideration are in direction N2 (by Theo-rem 6.10 this makes sense, since U2 2 Rpart). Let us prove that all theadditional units belong to RlocN1(U1) (see Fig. 8.1). Indeed, all such units ev-idently belong to RN1(U1). If U 2 RN1(U1) but U =2 RlocN1(U1), then �S(U)does not have edges in common with �S(U1) and lies behind N1 w.r.t. M1.However, by Theorem 7.3(iii), the part of �S(W ) lying behind N1 w.r.t.M1is deleted in the modi�ed carcass. Hence, b�S(W ) intersects b�S(U) at mostby a node, and thus U =2 bRloc. The backtracking rule in the cone RlocN1(U1)is justi�ed by Lemma 8.3. �8.3. Maintenance of the cell structure. As one can see easily fromthe description of the algorithm in Sect. 8.2, the most time-consuming prob-lem is to maintain the esh, namely, the distribution of the vertices of theinitial graph among units and reachability between units. Below we pro-pose a less detailed, cell oriented approach. We for sure keep track of thedistribution of vertices of G among cells, but do not guarantee the properdistribution of vertices among units. At any moment the partition of Vinto units represented by our data structure (henceforth, preunits) is, in aT

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63sense, intermediate: it is a re�nement of the true partition, while the initialpartition is a re�nement of this partition. More exactly, upon inserting anew edge we execute all the contractions as prescribed only if the resultingunit is a terminal; otherwise, we contract only the two preunits containingthe endpoints of the inserted edge, as described in Theorem 7.2(i). Thus,the pre esh can be not acyclic, but is a coherent locally orientable graph,and the true units are just its strongly connected components. Besides, wemaintain only the 2-partitions at the preunits, but not reachability cones ofany kind. The set of the preunits, the 2-partitions at each one of them, andthe skeleton are represented in the same way as in the previous algorithm.The preliminary stage of the new algorithm (distinguishing between trivialand nontrivial edge insertions) almost coincides with that of the previous oneand has the same complexity O(u). The only di�erence is that we locatenot the units, but the preunits Up1 and Up2 that contain the endpoints of thenew edge.The �rst stage of the new algorithm (update of the skeleton) coincidesliterally with that of the previous one and has the same complexity O(~n +~� log2 ~�).The only di�erence on the second stage (update of projections) is relatedto the case of units whose coordinates must be changed according to The-orem 7.3(iii). Instead of scanning the cones, we scan the two reachabilitysubgraphs of Up1 and Up2 . It is done by executing DFS four times (in bothdirections for both preunits) with backtracking each time when we reach apreunit whose projection no more intersects a certain path in the skeleton.It is easy to see that all the reasoning involving weights of edges remainsvalid in this case, hence the total amount of work on the second stage isO(~� ~m).The third stage of the new algorithm (update of the pre esh) is somewhatdi�erent from the corresponding stage of the previous one. If the new unit(and thus the new preunit) is not a terminal, we contract only the preunitsUp1 and Up2 . The 2-partition at the new preunit is maintained exactly as inthe previous algorithm with the help of labels, and the total amount of timefor this case is O(~n log ~n + ~� log2 ~�).If the new unit (and thus the new preunit) is a terminal, we just contractall the preunits whose new projection coincides with the projection of thenew terminal (see the remark after Theorem 7.3). Thus, the total amountof time in this case is O(~�~n+ ~n log ~n+ ~� log2 ~�) = O(~�~n+ ~n log ~n).To �nd whether two vertices of G are separated by an S-mincut, we�nd the corresponding preunits (in O(1) worst-case time). If they coincide,then such a cut does not exist. Otherwise, we check their projections. Ifthe projections di�er, then such a cut exists and is a cell cut. To �nd itwe analyze the projections of all the preunits and verify the inclusion ofTheorem 6.7. This can be done in O(~n) amortized time by means of NCAqueries as in the previous algorithm.Tec

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64If the projections coincide, then the two preunits belong to the same non-terminal cell. In this case we do not know at once whether a cut in questionexists or not; thus we answer both queries simultaneously. According toLemma 6.13, the work to be done is split into two parts. One of them con-sists in �nding a certain cell cut and is done exactly in the same way asin the previous case and within the same time. The other part is to �nda certain reachability cone of one of our preunits. It is done by scanningthe reachability subgraph of this preunit. Since, unlike the previous algo-rithm, we have no special data structure supporting reachability cones, thisis done similarly to the scanning on the second stage. Thus, the time forboth queries in this case is O( ~m).Finally, to obtain the strip WS1;S2 of Theorem 6.4 we proceed exactly asin the previous algorithm and get this strip up to contractions of stronglyconnected components. To get the true strip, we replace the ordinary DFSused in the previous algorithm for obtaining orientations by the extendedDFS that also �nds and contracts strongly connected components. The totaltime remains the same.We thus get the following result.Theorem 8.5. The cell structure of the connectivity carcass of an arbitraryvertex subset S can be maintained in O(~� ~m + ~n log ~n + u + q ~m) time foran arbitrary sequence of u edge insertions preserving the value of �S andq queries Sep(u; v), Cut(u; v), Strip(u; v). Moreover, if at least one of thevertices u, v is in S, or u and v belong to distinct cells, then the querySep(u; v) can be answered in O(1) worst-case time, and the query Cut(u; v)in O(~n) amortized time. References[AHU] Aho, A. V,. Hopcroft, J. E., and Ullman, J. D., The Design and Analysis ofComputer Algorithms, Addison-Wesley, Reading, MA, 1976.[Be] Benczur, A., Augmenting undirected connectivity in ~O(n3) time, Proc. 26th Symp.on Theory of Computers (STOC'94), 1994, pp. 658{667.[Bi] Bixby, R.E., The minimum number of edges and vertices in a graph with edgeconnectivity n and m n-bonds, Networks 5 (1975), 253{298.[DKL] Dinic, E. A., Karzanov, A. V., and Lomonosov, M. V., On the structure of thesystem of minimum edge cuts in a graph, Studies in Discrete Optimization, A. A.Fridman (Ed.), Nauka, Moscow, 1976, pp. 290{306, (in Russian, for a review inEnglish see [NV]).[DN] Dinitz, Ye. and Nutov, Z., Cactus-tree type models for families of bisections of aset (1997), preprint.[DV1] Dinitz, Ye. and Vainshtein, A., The connectivity carcass of a vertex subset in agraph and its incremental maintenance, Proc. 26th Symp. on Theory of Computing(STOC'94), 1994, pp. 716{725.[DV2] Dinitz, Ye. and Vainshtein, A., Locally orientable graphs, cell structures, and anew incremental algorithm for maintaining the connectivity carcass in a graph,Proc. 6th ACM{SIAM Symp. Discrete Algorithms (SODA'95), 1995, pp. 302{311.[DV3] Dinitz, Ye. and Vainshtein, A., The general structure of edge-connectivity of avertex subset in a graph and its incremental maintenance. Even case (1997), inpreparation.[DW] Dinitz, Ye. and Westbrook, J., Maintaining the classes of 4-edge-connectivity ina graph on-line (1997), accepted to Algorithmica.Tec

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65[FF] Ford, L.R.,Jr and Fulkerson, D.R., Flows in Networks, Princeton Un-ty Press,Princeton, NJ, 1962.[GI] Galil, Z. and Italiano, G. F., Maintaining the 3-edge-connected components of agraph on-line, SIAM J. Comput. 22 (1993), 11{28.[GH] Gomory, R.E., Hu, T.C., Multy-terminal network ows, SIAM J.Appl.Math. 9(1961), 551{560.[GN] Gus�eld, D. and Naor, D., Extracting maximal information about sets of minimumcuts, Algorithmica 10 (1993), 64{89.[I] Italiano, G. F., Amortized e�ciency of a path retrieval data structure, Theor.Comp. Sci. 48 (1986), 273{281.[LL] La Poutr�e J. and Leeuwen, J. van, Maintenance of transitive closure and transitivereduction of graphs, Proc. Workshop on Graph-Theoretics Concepts in ComputerScience, vol. 314, Springer-Verlag, Berlin, 1988, pp. 106-120.[N] Naor, D., Private communication (1991).[NGM] Naor, D., Gus�eld, D., and Martel, C., A fast algorithm for optimally increasingthe edge-connectivity, SIAM J. Comput. 26 (1997), 1139{1165.[NV] Naor, D. and Vazirani, V., Representing and enumerating edge-connectivity cuts inRNC, Proc. 2nd Workshop on Algorithms and Data Structures (WADS'91), Lec-ture Notes in Computer Science, vol. 519, Springer-Verlag, Berlin, 1991, pp. 273{285.[PQ] Picard, J.C. and Queyranne, M., On the structure of all minimum cuts in anetwork and applications, Math. Programming Study 13 (1980), 8-16.[W92] Westbrook, J., Fast incremental planarity testing, Proc. Int. Symp. on Automata,Languages and Programming (ICALP'92), Lect. Notes Comp. Sci., Springer-Verlag,NY, 1992.[W93] Westbrook, J., Incremental algorithms for four-edge connectivity., The abstractof the lecture to be held at the CS seminar of Bell Labs on March 10, 1993.[WT] Westbrook, J. and Tarjan, R.E., Maintaining bridge-connected and biconnectedcomponents on-line, Algorithmica 7 (1992), 433{464.T

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