A Mengerian Theorem for Paths of Length at Least Three

Michael Hager TECHNlSCHE UNlVfRSlTf T 6 fRL lN

1000 BERLIN 12 WEST GERMANY

ABSTRACT

Let be x, y two vertices of a graph G, such that t openly disjoint xy-paths of length 2 3 exist. In this article we show that then there exists a set S of cardinality less than or equal to 3 t - 2, resp. 2 t for t E {1,2,3}, which destroys all xy-paths of length 23. Also a lower bound for the car- dinality of S is given by constructing special graphs.

1. INTRODUCTION AND NOTATIONS

L. LovBsz, V. Neumann-Lara, and M. D. Plummer [3] studied Mengerian theo- rems for paths of bounded length and gave bounds €or the minimum size of a set T of points in a graph G, such that T destroys all paths with length less than or equal to a constant k between two vertices x , y of G. This bounds are given in terms of the maximum number of independent paths from x to y whose length does not exceed k .

In a recent article S . M. Boyles and G. Exoo [ l J gave a better lower bound by constructing a family of extreme graphs.

L. Montejano and V. Neumann-Lara [6] raised the question whether such variations of Menger’s theorem could be found for rhe dual problem, i.e., destroying all long paths between two vertices, and they gave a lower bound.

In this paper we will mainly restrict ourselves to the case n = 3 and follow Harary’s notation [ 2 ] .

Let be G a graph, S C V ( G ) and x , y Q V(G) - S. Then we write (x ,S,y); (n z- 2) if and only if every xy-path in G of lengrh greater or q u a 1 n has at least one point in S. If a E R, {a} will denote the minimum integer greater or equal than a.

L. Montejano and V. Neumann-Lara (161, Theorem 1) proved the following theorem: Journal of Graph Theory, Vol. 10 (1986) 533-540 Q 1986 by John Wiley & Sons, Inc. CCC 0364-9024/86/040533-08$04.00

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“Let n 2 2. If ( x , S , y ) g implies IS1 2 h then there exist {h/(3n - 5 ) ) openly disjoint xy-paths in G of length greater or equal n.”

For n = 3, the case n = 2 is Menger’s theorem, the existence of {h/4} xy- paths follows.

This means: If t openly disjoint paths of length 2 3 cxist, there is a set S C V ( G ) of cardinality less than or equal 4t with (x, S, y):.

Here we will give the following bounds: A) There exist graphs G, ( t E N), x , y E V(G,), such that t openly disjoint

xy-paths of length 2 n exist and each set S with (x, S, y):, has cardinality at least t (n - 1) (i.e., 2t for n = 3).

B) For each graph G, x , y E V ( G ) , with t openly disjoint xy-paths of length 23 , there is a set S C V ( G ) with (x, S, y); of cardinality less than or equal 32 - 2, t 2 4, and 2t, t E {1,2,3).

2. CONSTRUCTION OF EXAMPLES FOR A LOWER BOUND

Let be S , , . . . , S, t disjoint copies of the complete graph K2n-3 . Then we define the graph G, by:

Thus exact t openly disjoint xy-paths of length In exist and each set S C V(G,), which destroys all such paths, must contain n - 1 vertices of each S,. Hence IS1 2 t (n - 1).

Therefore the bound in Theorem 1 in [ 6 ] can be strengthened to at most {h l (n - 1)).

3. THE THEOREM

For the following let be G a connected graph and x , y E V(G) .

Theorem 1. Each graph G with at most t independent xy-paths of length 2 3 contains a vertex set S with IS1 5 3t - 2 , if t 2 4 , resp. for t E {1,2,3}, IS1 5 2t , such that ( x ,S ,y ) i .

Before we will prove Theorem 1, we formulate an auxiliary result. NG(z ) ) . Then we choose % out of the set of

all maximal systems of t independent xy-paths P,, 1 5 i 5 t , of length 2 3 Let be M = {z E V(G) 1 {x, y }

MENGERIAN THEOREM FOR PATHS 535

with minimal cardinality of M n V(%), such that the paths have minimal length. Let be M * = M - (M n V(%)).

Hence each path P I E 011 contains at most two points of M , and if m E M is a vertex of P I , then xm or my E E ( P , ) .

Now we change G into G(%), such that each edge k = xm, ym is dropped, if k does not belong to E(%).

Let be R the intersection of M with the neighborhood of x in G(Q), ( R ( = r, then we can formulate the following essential Lemma.

Lemma. There exists a set T 1 R with cardinality t which separates x and y in G(%).

Proof. Let be G ' = G(%) - R . Then an x - y separating set A in G ' exists with \A1 = a I t and G ' contains a system W of independent xy-paths with A C V(W). Let us assume that u > t - r for all such sets, otherwise R U A = T is the desired set.

Hence there exists at least one vertex z, E A n V ( P , ) for each path P , with P , r l V(G*) # $I, where G * is the x-component of G ' - A .

(I) There exists an xy-path-system W in G ' with cardinality u such that each path P, with V(W) fl P, # $I fulfills P, n R = pl. Here is stated a contradic- tion to the maximality of t .

(11) Each system W meets a path PI with P, n R # (3.

(a) P, n (A U V ( G * ) ) # $I Thus we find a new path-system %' with IM n V(%')I < IM fl V(%)l by 011' = (W n G*) U (Q - G*) (see Fig. 1). This is a contradiction to the mini- mality of (M n V(%)l.

(b) Each such path P, meets W behind A . Then we can change W to a system W ' , such that W' meets exactly ]A( paths

of %. For that let us choose the set X C_ V(W n %) of those vertices x , E V ( P , ) , which have minimum distance to y. Thus (XI 2 a.

If 1x1 > a, then a path W E W meets two different paths P I , PI, at xJ ,xt , say at first P,.

Now we change W to W , by W , = (W - W ) U W , , where W1 consists of W from x to x, and PI from x, to y. So a new set IX,l < 1x1 can be found and the same method may be used until IX*l = a is reached.

Thus a system W' with the desired property exists. Hence we can change % to a system 011' which contradicts the minimality of

M n V(011). We get %' by: W E %' if

- W = P k with P k f l W' = $I or

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A

C ( U ) w i t h C * a s a subgrapl i .

FIG. 1

- W contains the first part of W' E W' from x to x, E X * and the part of Pi from x, to y . I

Proof of Theorem 1. The proof is divided into three parts. First we show, using our Lemma, that there exists an x - y separating set S (in the sense of ( x , S , y ) i ) with cardinality at most 3 t . Second we prove that in the cases IS1 E {3 t , 3 t - 1) a separating set with cardinality 3 t - 2 can be found, and at last we have a look at the case IS1 = 7.

(I) We know that no paths Q with V ( Q ) n V(%) = 8 between vertices of M * exist, because otherwise at least t + 1 openly disjoint xy-paths can be found.

Now all paths Q = (y , , . . . ,yn) with y I , . . . ,yn-l q! V(%), which link M* to a certain path P E %, either have a common vertex m(P) E P - { x , y } or they begin at the same vertex m * ( P ) E M*. Otherwise also more paths exist.

Hence we choose far each P E % the vertex m*(P) or, if it does not exist, m(P) and get the set Y . Then S = Y U T U ( M fl (V(%) - T ) ) , T chosen as in the Lemma, is a set of cardinality at most 3 t , which destroys all xy-paths of length 2 3 .

(11) (a) JSI = 3r Thus IM n V(%)l = 2t and for each two paths P , P' the vertices m*(P) have to be different, If they are used in S. Otherwise IS/ I 31 - 1.

Therefore paths between different paths P i , PI only can exist, if they meet m(Pi) or m(Pj) . Otherwise a path-system '%' with more paths exist (see Fig. 2).

Hence S' = T U Y' destroys all xy-paths of length 2 3 , if we choose Y ' as the union of the vertices m(P) and m * ( P ) , if m(P) does not exist.

MENGERIAN THEOREM FOR PATHS 537

U'

X Y

FIG. 2

Here S' is a separating set of cardinality at most 2t.

(b) (St = 3t - 1. Let be IYI = t. In this case an argument similar to that in (a) holds, because for each two paths P I , P, there is one with a vertex m E M - T; notice that the argument is symmetric with x and y. Therefore a separating set with at most 2t elements exists.

Hence IYI = t - 1 . Let us assume that only one path P I has no vertex m*(P1) or m(P1) .

If P I is isolated, then S - {m} is a separating set with 3r - 2 elements (see Fig. 3). Therefore let be F a path between PI and Pk and F does not meet m(Pk), otherwise s - {m} separates x and y , too. Hence (% - u F k (see Fig. 3) is a new system with IYI = r, if m(Pk) belongs to Y , or for all paths m * ( P I ) or m ( P , ) exist. Thus two paths P , , P , exist with m * ( P I ) = m*(P,).

FIG. 3

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Hence S - {m*(P;)} is a separating set with 3t - 2 elements, because each xy-path which avoids T U (M - M*) has to meet M * two times. (Notice that (M n v(%)l = 21.)

(111) At this point we have proved that each graph with t openly disjoint xy-paths of length 2 3 contains a set S of cardinality less than or equal to max(3t - 2 , 2 t ) with (x ,S ,y ) i .

So for t = 1, IS1 5 2t and for 2 5 t , IS] I 3t - 2 holds. If t = 2, then 2t = 3t - 2, and if t = 3, then 2t = 6 < 7 = 3t - 2.

Hence we only have to show that, if there are three paths and a separating set of seven vertices, chosen using case (II)(b), there exists a separating set with six vertices, too.

(a) I Y I = 3. In this case there is a separating set with six vertices, if not IM n V(%)l = 2, and the situation of Figure 4 holds (see (II)(b)) otherwise a path-system with at least four xy-paths exists. Let be m l , m2 E M n V(%), xl, x2 the groundvenices with respect to ml*, m2*, which have the smallest distance to x on PI, P 2 . z I , z2 are groundvertices of connecting paths P between PI and P 2 , which have smallest distance to x l , x 2 .

Now we choose a set T’ 3 m2 such that z l , z 2 are separated from xl ,x2 (IT’I = 3). This is possible, because if T’ does not exist, then, for example, a circuit C through u l , z l , x l , y l exists, which does not meet V ( P z ) U V ( P 3 ) U M * U {mI}. Hence four paths from x to y of length 2-3 exist; for example (x, . . . , U I , . . . , y l , . . . , y ) , ( x , . . . , Z Z , z I , . . . , x l , . . . , m l * , y ) , ( x , m 2 * . x2 , . . . ,y), P 3 , if C = ( z l r u I , . . . ,y l ,x , , . . . , zI) . For other types of C equiva- lent path-systems can be found. The same holds for P 2 . Therefore T ’ exists and T’ U Y is a separating set.

FIG. 4

MENGERIAN T H E O R E M FOR PATHS 539

(b) IYI = 1. Here S - {m} with m E M n V(%) and my E E(%) separates .r and y , beP40 cause two elements of (M fl V(Q)) - T must be used to get a path of length 2 3 , which does not meet T U M*.

(c) (YI = 2. Here at least m l , m 2 E (M n V(%)) - T exist, the existence of a vertex m3 is negligible.

Let us assume that there is a path PI such that m*(P1) and m(Pl) do not exist. Now paths between P2 and P3, which do not meet m ( P 2 ) or m(P,), can-

If m l E P I , then S - {m2 ,m3} separates. If ml @ P I , then we choose the vertex zI, which is the endvertex of a path

between PI and P2 resp. P3 on P , , which has the smallest distance to y and its neighbor uI on PI in direction to x . Thus there is no circuit in G - (PI U P2 U M ) with u I , z I , u on it (see (IIl)(a)). Therefore a vertex z separates z l and y in G - ( P I U P2 U M ) . Hence T U Y U { z } is a separating set of cardinality 6. Otherwise m * ( P I ) = m * ( P 2 ) .

If P3 is isolated from PI U P2, then a six-point-separating set can be found at once. Hence at least one path between PI U P2 and P3 exists. But this path must meet one groundvertex m(P,) , otherwise a four-path-system exists (notice that x and y are symmetric). Following the same argument, no path between PI and P2 exists which avoids m ( P , ) . Thus T U Y’, where Y ’ is the union of the single groundvertices m(P,) or, if m(P,) does not exist, of m * ( P I ) , has the desired property. I

not exist.

The following Theorem is an immediate consequence of Theorem I

Theorem 2.

h E { 1, . . . ,8}, openly disjoint xy-paths in G of length 2 3 .

Let G be a graph, x , y E V ( G ) and S C V ( G ) - { x , y } . If ( x , S , y ) ? implies IS1 2 h , then there exist {(h + 2) /3} , resp. {h /2} for

Corollary. For h E { 1, . . . ,8}, {h /2} is a sharp bound (see Section 2).

At last we state two conjectures. Notice that Conjecture 1 is a special case of Conjecture 2.

Conjecture 1. {h /2} is a sharp bound for each h in the case n = 3.*

Conjecture 2. {h/(n - 1)) is a sharp bound for each h for an arbitrary n.

*In the meantime since this paper was written W. Mader 151 proved Conjec- ture 1 using results of 141.

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References

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[2] F. Harary, Graph-Theory. Addison-Wesley, Reading, MA (1969). [3] L. Lovlsz, V. Neumann-Lara, and M. D. Plummer, Mengerian theorems

for paths of bounded length. Period. Math. Hungar. 9 (1978) 269-276. 141 W. Mader, Ubcr die Maximalzahl kreuzungsfreier H-Wege. Arch. Math. 31

[5] W. Mader, On disjoint paths in graphs. Preprint (1985). [6] L. Montejano and V. Neumann-Lara, A variation of Menger’s theorem for

bounded length. J . Graph Theory 6 (1982) 205-209.

(1978) 387-402.

long paths. J . Comb. Theory, Series B 36 (1984) 213-217.