Dominating Sets with SmallClique Covering Number

Stephen G. Penrice*DEPARTMENT OF MATHEMATICS

SUNY COLLEGE AT CORTLANDCORTLAND, NEW YORK 13045

E-mail: spenrice@ets.org

Received November 10, 1994; revised January 1, 1997

Abstract: Motivated by earlier work on dominating cliques, we show that if a graphG is connected and contains no induced subgraph isomorphic to P6 or Ht (the graphobtained by subdividing each edge of K1,t, t ≥ 3, by exactly one vertex), then G has adominating set which induces a connected graph with clique covering number at mostt− 1. c© 1997 John Wiley & Sons, Inc. J Graph Theory 25: 101–105, 1997

Keywords: dominating set, clique covering number

1. INTRODUCTION

We discuss only finite undirected graphs without loops or multiple edges. Let G = (V,E) be agraph. If x ∈ V , let N(x) denote the set of neighbors of x in G. Let N [x] denote {x} ∪N(x).For S ⊆ V , GS denotes the subgraph of G induced by S. A dominating set for G is a set S ⊆ Vsuch that if x ∈ V −S, then N(x)∩S 6= ∅. Let θ(GS) denote the clique covering number of GS ,i.e., the smallest integer k such that S can be partitioned into k subsets, each of which inducesa clique. Let α(GS) denote the size of the largest independent set in GS . If G1, G2, . . . , Gm

are finite graphs, Forb(G1, G2, . . . , Gm) denotes the class of graphs G such that G contains noinduced subgraph isomorphic to Gi for all i, 1 ≤ i ≤ m. We use Pn(Cn) to denote the path(cycle) on n vertices.

* Received support as a DIMACS Research/Education Postdoctoral Fellow from Aug. 1, 1994to July 31, 1995. Funding for this position was provided through NSF contract STC 91-19999.Current address: Assessment Division, Educational Testing Service, Princeton, NJ 08541.

c© 1997 John Wiley & Sons, Inc. CCC 0364-9024/97/020101-05

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Our work is motivated by the following results of Bacso and Tuza [1], the first of which wasproved independently by Cozzens and Kelleher [5]:

Theorem 1. If G ∈ Forb(P5, C5) is connected, then G has a dominating set D such that GD isa clique.

Theorem 2. If G ∈ Forb(P5) is connected, then G contains a dominating set D such that GD

is either a clique or a P3.

In discussing their motivation, Cozzens and Kelleher mention that in a communications networkit may be desirable to have a core of nodes such that any two nodes in the core can communicate,and every node in the network communicates with some node of the core. If such a core does notexist, a small number of cliques, which collectively communicate with every node of the network,could serve a similar purpose. For instance, a research community may wish to organize itselfaround a few groups of researchers who share common interests and who are positioned withinthe community so that every member of the community can learn the results of at least one group.Thus, we will investigate conditions that guarantee that a graph G will have a dominating setwhich can be covered with a small number of cliques.

This paper discusses a result for some classes of graphs which properly contain Forb(P5).Note that forbidding long paths is reasonable, since no positive result is possible for a class ofgraphs which includes Pk for arbitrarily large k. The next question one would naturally ask iswhether there is a constant c such that if G ∈ Forb(P6) is connected, G contains a dominating setD with θ(GD) ≤ c. Consider the graph Ht formed by subdividing each edge of K1,t by exactlyone vertex, where t is a positive integer. Clearly Ht ∈ Forb(P6), and any subgraph induced by adominating set of Ht contains an independent set of size t. Thus no positive result is possible forForb(P6), but we can show the following.

Theorem 3. Suppose t ≥ 3. If G ∈ Forb(P6, Ht) is connected, then G contains a dominatingset D such that GD is connected and θ(GD) ≤ t− 1.

Clearly this bound is best possible, since Ht−1 ∈ Forb(P6, Ht). We claim that no resultanalogous to Theorem 3 is possible for Forb(P7, H3); in fact, no positive result is possible forForb(P7, 3K2), where 3K2 denotes the graph consisting of 3 independent edges. LetM2, M3, . . .be a sequence of graphs such that, for all integers i ≥ 2, α(Mi) = 2 and θ(Mi) = i. Let Gi bethe graph obtained by attaching a vertex of degree 1 to each vertex of Mi. The reader may verifythat for all i, Gi ∈ Forb(P7, 3K2), and that if D is a dominating set for Gi, then the subgraphinduced by D has clique covering number ≥ i.

The example in the previous paragraph also helps to answer the following question: For whichfinite graphs Γ does there exist a constant cΓ such that if G ∈ Forb(Γ) is connected, G containsa dominating set D with θ(GD) ≤ cΓ? Clearly, if cΓ exists, Γ is an induced subgraph of C2cΓ+3,HcΓ+1, andGcΓ+1. This implies that Γ is isomorphic to an induced subgraph ofP5 + (cΓ − 2)K1.On the other hand, we have the following proposition.

Proposition 1. If m ≥ 2 and G ∈ Forb(P5 + (m− 2)K1), then G contains a dominating set Dsuch that θ(GD) ≤ m.

Proof. We use induction. The base casem = 2 follows directly from Theorem 2. Suppose thestatement holds for m = k. Let G ∈ Forb(P5 +(k−1)K1) be connected. If G ∈ Forb(P5 +(k−2)K1) then, by the induction hypothesis,G contains a dominating setD such that θ(GD) ≤ k, andwe have the desired result for m = k+1. Otherwise G contains an induced subgraph isomorphicto P5 + (k − 2)K1, and since G ∈Forb(P5 + (k − 1)K1), this subgraph is a dominating set

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for G. Clearly α(P5 + (k − 2)K1) = k + 1, so we have the result for m = k + 1 in this caseas well.

At least two other directions suggested by Theorem 1 have been studied. Bacso and Tuza haveaddressed the question of finding dominating sets with small diameter [2, 3, 4]. In particular, in[2] they show that if G ∈ Forb(Pk) is connected, it contains a dominating set D such that GD

has diameter at most k − 3. The author [6] has written about bounding the domination numberfrom above with a function of the clique size, showing that for all positive integers k and t, thereexists a function fk,t such that if G ∈ Forb(Pk, Ht) is connected, then G contains a dominatingset D with |D| ≤ fk,t(ω(G)).

2. PROOF OF THE THEOREM

In this section we prove Theorem 3. The structure of the proof is modeled after the proof ofTheorem 2 in [1]. One of the most powerful tools from that proof is the following lemma. If Gis a graph and S is a cutset in V (G) such that S ⊆ N [x] for some x ∈ S, then S is a star-cutset.

Lemma 1. Let D be a class of connected graphs. Let G be a minimal connected graph suchthat there is no dominating set D for G with GD ∈ D. Then either G has a cutvertex or G hasno star-cutset.

Taking D to be the class of connected graphs H with θ(H) ≤ t−1, we need to prove Theorem3 only for the two cases mentioned in Lemma 1. Lemma 2 will help us with these two cases.

If G ∈ Forb(P6, Ht), we say that v ∈ V (G) is critical if either v is the endpoint of a chordlessP5 in G, or v ∈ S ⊆ V (G) such that S − {v} induces Ht−1 and v's only neighbor in S isthe central vertex of the induced Ht−1. In other words, v is critical if v belongs to an inducedsubgraph of G in which attaching a leaf to v gives a graph isomorphic to either P6 or Ht. Themotivation for this definition is the fact that a cutvertex of G ∈ Forb(P6, Ht) is not critical.

Lemma 2. Suppose t ≥ 2. Let G ∈ Forb(P6, Ht) be connected. Suppose there is a vertexv ∈ V (G) such that v is not critical. Then G has a dominating set D such that v ∈ D, GD isconnected, GD contains no chordless P4's and θ(GD) ≤ t− 1.

Proof. We apply induction on the number of vertices of G. The result is trivial for a graphwith one vertex. Suppose thatG ∈ Forb(P6, Ht) is connected and hasn vertices, that v ∈ V (G) isnot critical, and that the result holds for all connected graphs in Forb(P6, Ht) with n−1 vertices.If v has a neighbor x which is not a cutvertex of G, then by the induction hypothesis G − {x}has a dominating set D with the desired properties. Since v ∈ D, D is a dominating set for Gas well.

Assume henceforth that every neighbor of v is a cutvertex.Since v is not critical, the set {x ∈ V (G): d(x, v) ≤ 2} is a dominating set forGwhich induces

a connected subgraph in G. Let N2 be a minimal subset of {x ∈ V (G) : d(x, v) = 2} such thatD = {v} ∪N(v)∪N2 is a dominating set for G. We claim that D is the desired dominating set.Clearly, v ∈ D, and GD is connected. We first show that α(GD) ≤ t − 1. Then we will arguethat GD does not contain a chordless P4. The result will follow since P4-free graphs are perfect.

First we observe that for every vertex r ∈ D − {v} there exists a vertex, say f(r), such thatf(r) ∈ {V (G)−D} and N(f(r)) ∩D = {r}. If r ∈ N(v), f(r) is any vertex that is not in thesame component of G− r as v; the existence of such a vertex follows from the assumption thatr is a cutvertex. If r ∈ N2, f(r) is any vertex that is not dominated by D− {r}; the existence ofsuch a vertex follows from the minimality of N2.

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Next we observe that if N2 6= ∅, then there is a vertex m in N(v) that is adjacent to all ofN2. Suppose a and b are vertices in N2. We claim that either N(a) ∩ N(v) ⊆ N(b) ∩ N(v)

or N(b) ∩ N(v) ⊆ N(a) ∩ N(v). If there exists c ∈ (N(a) ∩ N(v)) − (N(b) ∩ N(v)) andd ∈ (N(b) ∩ N(v)) − (N(a) ∩ N(v)), then a 6∼ b, or else {v, c, a, b, f(b)} induces a P5 withan endpoint at v, contradicting the hypothesis that v is not critical. Likewise, f(a) 6∼ f(b),or else {v, c, a, f(a), f(b)} induces a P5 with an endpoint at v. But then either c 6∼ d and{f(a), a, c, v, d, b, f(b)} induces P7, or c ∼ d and {f(a), a, c, d, b, f(b)} induces P6. We havea contradiction in either case, so the claim is proved. It follows from the claim that there is aneighbor m of v which is adjacent to all of N2.

Now suppose that D contains an independent set I with |I| = t. If I ∩ N2 = ∅, then{v}∪ I ∪{f(a): a ∈ I} induces Ht, a contradiction. So suppose I ∩N2 6= ∅, and let b ∈ I ∩N2.Then m 6∈ I . Moreover, m ∼ e for all e ∈ I ∩ N(v), for otherwise {f(e), e, v,m, b, f(b)}induces P6. Thus, if v ∈ I , then {m} ∪ I − {v} ∪ {f(a): a ∈ I} induces Ht−1, and therefore,since v ∼ m, v is critical, a contradiction. If v 6∈ I , then {m} ∪ I ∪ {f(a): a ∈ I} induces Ht, acontradiction. This completes the proof that α(GD) ≤ t− 1.

Now we claim that GD does not contain a chordless P4. Suppose P = {w, x, y, z} ⊆ D

induces a chordless P4 with endpoints w and z. If {w, z} ⊆ N(v), then w and z are cutverticesof G, and we may find vertices p and q such that {p, w, x, y, z, q} induces P6, a contradiction. Ifone endpoint of P is in N(v), the other endpoint cannot be v since v is not critical. Thus we mayassume without loss of generality that z ∈ N2. By the minimality of N2 we may find a vertexq ∈ V (G) −D such that N(q) ∩D = {z}, and thus {w, x, y, z, q} induces P5. Since v is notcritical,w 6= v. Ifw ∈ N(v), we may use the assumption thatw is a cutvertex ofG to find a vertexp such that {p, w, x, y, z, q} induces P6, a contradiction. If w ∈ N2, we may find p ∈ V (G)−D

such that N(p)∩D = {w}. Note that p 6∼ q, or else {q, p, w, u, v}, where u ∈ N(v) and u ∼ w,induces a chordless P5 with endpoint v, a contradiction. But then {p, w, x, y, z, q} induces P6, acontradiction. Thus no subset of D induces a chordless P4.

Since graphs which lack chordless P4's are known to be perfect, θ(GD) = α(GD) ≤ t − 1,and the lemma is proved.

Proof of Theorem 3. Let D be the class of connected graphs H with θ(H) ≤ t − 1. ByLemma 1 it suffices to prove the theorem for connected graphs in Forb(P6, Ht) which either havea cutvertex or do not have a star-cutset.

Suppose G ∈ Forb(P6, Ht) is connected and has a cutvertex v. Note that v is not critical. ByLemma 2, G has a dominating set D such that GD is connected and θ(GD) ≤ t− 1.

Suppose G ∈ Forb(P6, Ht) is connected and has no star-cutset. Choose any v ∈ V (G), andany u ∈ N(v). Let Y = {v} ∪ (V (G) − N [u]). By supposition, N [u] − {v} is not a cutset,so GY is connected. Moreover, v is not critical in GY , or else we could attach u to v to formeither P6 or Ht. Thus, the hypotheses of Lemma 2 are satisfied, and we may find a dominatingset DY for GY which has the properties given in the Lemma. If DY is a dominating set for G,then we are done, so assume there exists a vertex w ∈ N(u) such that N(w) ∩ DY = ∅. Weclaim that D′ = {u, v, w} ∪ (N(v) ∩ DY ) is a dominating set for G. Suppose x ∈ Y has noneighbor in D′. Then there exists a vertex y ∈ (DY − D′) ∩ N(x). Since GDY

is connectedand y ∈ (DY −D′), there exists a chordless path from y to v in GDY

of length at least 2. Butthen there exists a chordless path in G from x to w of length at least 5, a contradiction since thisimplies the existence of a chordless P6 in G. Hence, D′ is a dominating set for G.

Let N1 be a minimal subset of N(v) ∩DY such that D = {u, v, w} ∩N1 is a dominating setfor G. Clearly, GD is connected. We will show that θ(GD) ≤ t− 1.

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We claim that α(GN1) ≤ t − 2. Suppose {a1, . . . , at−1} ⊆ N1 is an independent set. Bythe minimality of N1, for each i, 1 ≤ i ≤ t − 1 there exists a vertex bi ∈ V (G) −D such thatN(bi) ∩D = {ai}. Note that {bi: 1 ≤ i ≤ t− 1} is independent; if, for example, b1 ∼ b2, then{b2, b1, a1, v, u, w} induces P6. But then {u, v, w}∪{ai : 1 ≤ i ≤ t− 1}∪{bi : 1 ≤ i ≤ t− 1}induces Ht, a contradiction.

By Lemma 2, GDYcontains no chordless P4. Therefore, GN1

contains no chordless P4, soθ(GN1) ≤ t− 2. If N1 = ∅, then the result is obvious because θ(G{u,v,w}) = 2 ≤ t− 1, sincet ≥ 3. Otherwise, we may add v to any one of the cliques in a minimum clique covering forGN1

to show that θ(GN1∪{v}) ≤ t− 2. Since {u,w} induces a clique, θ(GD) ≤ t− 1, and theTheorem is proved.

References

[1] G. Bacso and Z. Tuza, Dominating cliques in P5 -free graphs, Periodica Math. Hungar. 21 (1990),303–308.

[2] G. Bacso and Z. Tuza, A characterization of graphs without long induced paths, J. Graph Theory 14(1990), 455–464.

[3] G. Bacso and Z. Tuza, Dominating subgraphs of small diameter, submitted.

[4] G. Bacso and Z. Tuza, Domination properties and induced subgraphs, Discrete Math. 111 (1993),37–40.

[5] M. B. Cozzens and L. L. Kelleher, Dominating cliques in graphs, Discrete Math. 86 (1990), 101–116.

[6] S. Penrice, Upper bounds on domination number in terms of clique size, preprint.