Forbidden OrderedSubgraph vs. ForbiddenSubgraph Characterizationsof Graph Classes

Mark Ginn*DEPT. OF MATHEMATICAL SCIENCES

APPALACHIAN STATE UNIVERSITYBOONE, NC 28608

E-mail: mcg@math.appstate.edu

Received September 25, 1996; revised August 7, 1998

Abstract: We show that the minimum set of unordered graphs that must beforbidden to get the same graph class characterized by forbidding a single orderedgraph is infinite. c© 1999 John Wiley & Sons, Inc. J Graph Theory 30: 71–76, 1999

Keywords: forbidden subgraph, forbidden ordered subgraph, graph characterizations

1. INTRODUCTION

Two natural ways of characterizing classes of graphs are through the use of forbid-den ordered subgraphs, and forbidden subgraphs. In this article, we will explorethe relationship between these two characterizations.

For an ordered graph (G, <), that is, a finite graph G along with a linear order <on its vertex set V (G), we define the class Forb(G, <) to be the set of all graphs

*This work was done while the author was at Dept. of Mathematics and ComputerScience, Austin Peay State University, Clarksville, TN 37044.

c© 1999 John Wiley & Sons, Inc. CCC 0364-9024/99/020071-06

72 JOURNAL OF GRAPH THEORY

Γ for which there exists a linear order <′ of V (Γ) such that (Γ, <′) contains noinduced subgraph isomorphic to (G, <). By an ordered graph isomorphism wemean an order preserving as well as edge preserving bijection between the vertexsets. Many classes of graphs have been characterized as Forb(G, <) classes forthe proper selection of (G, <). For more on this see [1] or [2].

For a given set of graphs G we define the class Forb(G) to be the set of graphsthat do not contain any graph H ∈ G as an induced subgraph. This is another oftenused characterization of graph classes. See [5] for more on these.

From the following remark of Greenwell, Hemminger, and Kleirlein [5] onecan deduce that any Forb(G, <) class is also a Forb(G) class for some set ofgraphs G.

Remark 1. A class of graphs G can be characterized as a Forb(G) class forsome set of graphs G if and only if G is closed under taking induced subgraphs.

A moments thought will show that Forb(G, <) classes are closed under takinginduced subgraphs. For a given ordered graph (G, <) we will denote by G(G,<) theminimum set of unordered graphs giving the relation Forb(G, <) = Forb(G(G,<)).Thinking of the poset of all graphs partially ordered by induced containment, theset G(G,<) is the set of minimum elements in the complement of Forb(G, <) and,hence, is well defined.

Our goal here is to explore the size of G(G,<) for various ordered graphs (G, <).In [2] we showed that the problem of determining if a graph Γ belonged to theclass Forb(G, <) is NP complete for almost all graphs G that are 2-connected orhave 2-connected complement. This would certainly seem to imply that G(G,<) isinfinite for all these ordered graphs. In fact, much more is true as the followingtheorem states.

Theorem 1. For any ordered graph (G, <) where G is neither a complete graphnor an empty graph, G(G,<) is infinite.

Our purpose for the rest of this article is to prove this result.

2. PRELIMINARIES

We begin with a few preliminary remarks and a lemma.First, note that our condition that (G, <) is neither complete nor empty is nec-

essary as G(G,<) = {G} if (G, <) is either a complete graph or an empty graph.The next two remarks follow directly from the definitions given above.

Remark 2. If (H, <) is an induced subgraph of (G, <), then

Forb(H, <) ⊆ Forb(G, <).

Remark 3. For any ordered graph (G, <),G(Gc,<) = Gc(G,<).

By Gc we mean the complement of the graph G, and by Gc(G,<) we mean the set

of graphs consisting of the complements of each of the graphs in G(G,<). Note that

CHARACTERIZATIONS OF GRAPH CLASSES 73

this remark implies that we can always assume that our ordered graph (G, <) isconnected, because if it is not, we can consider its complement, which is connected,and the same number of unordered graphs must be forbidden for either graph.

The following technical lemma is a key tool in our proofs. This lemma may befound in [6] and is essentially the same as a lemma found in [3] or [4]. We includean outline of the proof for completeness.

Given a hypergraph H(V, E), a cycle in H of length n is a sequence

(e1, v1, e2, v2, . . . , en, vn),

where ei ∈ E, vi ∈ V, vi ∈ ei ∩ ei+1 : 1 ≤ i ≤ n − 1, and vn ∈ e1 ∩ en. The girthof H is the length of the shortest cycle in H. If H contains no cycles, we say it hasinfinite girth.

Lemma 1. Given positive integers n and k, let ε = 1k . There exists an integer

N1 = N1(n, k) such that for all N ≥ N1 there exists an n uniform hypergraphHN with the following properties:

1. |V (HN )| = N.2. |E(HN )| > N1+ε.3. girth(HN ) > k.

Proof. Consider the set H(N, m) of all n-uniform hypergraphs (V, E), where|V | = N and |E| = m = 2dN1+εe.

Then the average number of edges contained in a cycle of length at most k isless than

k∑j=1

c(n, j)(

N(n − 1)j

)(

(Nn ) − jm − j

)(

(Nn )m

) = O(N),

where c(n, j) > 0 is a function of n and j, which does not depend on N .Hence, for N sufficiently large there exists a hypergraph H ∈ H(N, m) such

that H contains at most N1+ε edges in cycles of length at most k. After deletingthese edges we are left with a hypergraph HN satisfying the conditions of ourlemma.

3. PROOF OF THEOREM 1

We begin by developing some terminology that will prove useful. Given two graphsG and H , with |V (G)| = n, we define the n-uniform hypergraph HG(H) on V (H)to have edge set E ⊆ [V ]n such that e ∈ E if and only if e is isomorphic to G as aninduced subgraph of H . We also define the hypergraph HC(H) to have vertex setV = V (H) and edge set E ⊆ P(V ) such that e ∈ E if and only if e is containedin some edge of HG(H) and is a clique when considered as a subgraph of thatcopy of G. Here we are using the more restrictive definition of a clique to be both

74 JOURNAL OF GRAPH THEORY

complete and maximal. We will ignore duplicate edges arising from different edgesof HG(H) and, if one edge of HC(H) contains another edge (obviously from adifferent edge of HG(H)), we will delete the smaller edge from HC(H). Note thatHC(G) is the hypergraph on the vertex set of G whose edges induce the cliques ofG. The following lemma follows easily from these definitions and will be exploitedoften in our proof.

Lemma 2. For any graph H and any ordered graph (G, <) that is neithercomplete nor empty, if H /∈ Forb(G, <) then both HG(H) and HC(H) havefinite girth.

The proof of Theorem 1 will be broken down into two cases. First we will showthat the theorem is true for those graphs G such that HC(G) has infinite girth andthen will look at those graphs where this is not the case.

3.1. Case 1: HCCC(GGG) Has Infinite Girth

Proposition 1. If G is a connected graph such that HC(G) has infinite girth and< is any ordering of V (G), then G(G,<) is infinite.

Proof. Suppose that G(G,<) is finite. Let k = maxX∈G(G,<){girth(HC(X)}.Note that k must be finite, since by Lemma 2 HC(X) must contain a cycle forall X ∈ G(G,<). Our strategy will be to construct a graph Γ /∈ Forb(G, <)such that girth(HC(Γ)) > k. Our proposition follows, since this implies Xis not an induced subgraph of Γ for all X ∈ G(G,<) contradicting the fact thatForb(G, <) = Forb(G(G,<)).

Let n = |V (G)| and HN be the n-uniform hypergraph promised by Lemma 1with girth greater than max{k, 3}, |V (HN )| = N , and |E(HN )| > N1+ε. Letl = n!

|Aut(G,<)| > 1. There are l order nonisomorphic ways of placing edges onan ordered vertex set (1, 2, . . . , n) to get a graph isomorphic to G. Consider arandom insertion of copies of G into the edges of HN with uniform probability andindependence between insertions. (This is possible, since no two edges share morethan a single vertex.) For any ordering <′ of V (HN ) and any e ∈ E(HN ), let A<′

e

be the event that (e, <′) is not isomorphic to (G, <). Then,

Pr(A<′e ) =

l − 1l

.

If we let A<′be the event that no edge of HN contains an ordered subgraph not

isomorphic to (G, <) (A<′= ∩e∈E(HN )A

<′e ). Then

Pr(A<′) =

(l − 1

l

)N1+ε

.

Finally, if A is the event, where exists an ordering <′ of HN such that A<′holds,

Pr(A) = N !(

l − 1l

)N1+ε

= o(1).

CHARACTERIZATIONS OF GRAPH CLASSES 75

Note, that Ac is the event, where every ordering of V (HN ) contains a hyperedgecontaining a subgraph isomorphic to (G, <) and Pr(Ac) > 0 for N large enough.Hence, there is a way of inserting copies of G into the edges of HN (for largeenough N ) to form a graph X such that X /∈ Forb(G, <).

Since girth(HN ) > 3 and every edge of X was inserted into an edge of HN ,all cliques in X must be contained within a single edge of HN . Hence, all edgesof HC(X) are contained in edges of HN . Since HC(G) has infinite girth andevery edge of HN contains a copy of G, any cycle in HC(X) is contained in a(possibly shorter) cycle of HN . This implies girth(HC(X)) ≥ girth(HN ) > k,thus completing our proof.

3.2. Case 2: Other Graphs

If HC(G) has finite girth, then there is no way to construct a graph X /∈ Forb(G, <)such that HC(X) has arbitrarily large girth, since X must contain many copies ofG that will force the girth of HC(X) to be no bigger than that of HC(G). However,if HC(G) has finite girth, then G contains a 2-connected noncomplete subgraph(just consider the subgraph induced by the vertices in some cycle of HC(G)). Thisfact will allow us to complete our proof.

Proposition 2. For any graph G that contains a 2-connected noncomplete sub-graph H (possibly equal to G itself) and any ordering < of V (G),G(G,<) is infinite.

Proof. Let (H, <) be a maximal 2-connected noncomplete subgraph of (G, <).The maximality of H implies that HH(G) has infinite girth. Also, by Remark2, G(G,<) ∩ Forb(H, <) = ∅ so, for any graph X ∈ G(G,<),HH(X) has finitegirth. Our proof now proceeds in the same manner as the proof of Proposition 1.Assume that G(G,<) is finite. Let k = maxX∈G(G,<){girth(HH(X)}. Construct ann-uniform hypergraph HN such that girth(HN ) > max{k, l}, where l = |V (H)|.Insert copies of G into the edges of HN like before to form a graph Γ /∈ Forb(G, <).Since H is 2-connected, the girth of HN being larger than |V (H)| forces every copyof H in Γ to be in an edge of HN . Thus,

girth(HH(Γ)) > girth(HN ) > k.

This implies X /≤ Γ for all X ∈ G(G,<), contradicting the fact that Forb(G, <) =Forb(G(G,<)). The details are analogous to the previous case and are left to thereader.

In review, we have shown that, for any connected graph G that is not complete, Gsatisfies the hypothesis of either Proposition 1 or 2 and, hence, that for any ordering< of V (G), G(G,<) is infinite. Since any graph or its complement is connected, thiscompletes the proof of our theorem.

76 JOURNAL OF GRAPH THEORY

4. CONCLUSION

There are a couple of other interesting related questions that one could ask.First, in [2] we showed that for almost all 2-connected ordered graphs (G, <), the

question of determining if a graph is in Forb(G, <) is NP complete. Conversely,there are some classes of graphs including comparability and chordal graphs thathave Forb(G, <) characterizations for which efficient recognition algorithms areknown. This certainly seems to imply a simpler structure of G(G,<) in the lattercase. Is there, in fact, some distinguishing feature of these G(G,<) sets for easier-to-recognize classes?

Second, there is a natural generalization of the Forb(G, <) characterization toa Forb(G, <) characterization, where instead of asking if there is an ordering ofthe vertices of a graph that forbids a single ordered subgraph, we ask if there is onethat forbids a whole set (G, <) of ordered subgraphs. What can be said about thesize and or structure of the sets G(G,<)? The same techniques used in this articleshow that, if the set (G, <) is made up only of different orderings of the same graphG, then |G(G,<)| = 1 if (G, <) contains all orderings of G; and G(G,<) is infinite,if (G, <) does not contain all of these orderings. This, however, leaves open theoften used characterizations where (G, <) contains different graphs.

References

[1] P. Damaschke, ‘‘Forbidden ordered subgraphs,’’ Topics in combinatorics andgraph theory, R. Bodendiek and R. Henn, (Editors), Physica–Verlag, Heidel-berg, 1990, pp. 219–229.

[2] D. Duffus, M. Ginn, and V. Rodl, On the computational complexity of orderedsubgraph recognition. Random Struct Algo 7 (1995), 223–267.

[3] P. Erdos, Graph theory and probability. Can J Math 11 (1959), 34–38.[4] P. Erdos, A. Hajnal, On chromatic number of graphs and set systems. Acta

Math Acad Sci Hungar 17 (1966), 61–99.[5] D. L. Greenwell, R. L. Hemminger, and J. Kleirlein, Forbidden subgraphs,

Proc 4th SE Conf Combin Graph Theory Comp 1973, pp. 389–394.[6] J. Nesetril and V. Rodl, On a probabilistic graph-theoretical method, Proc

AMS 72, 1978, 417–421.