COMPOSITION OF STABLE SET POLYHEDRA Benjamin McClosky and Illya V. Hicks Department of Comptational and Applied Mathematics Rice University November 30, 2007 Abstract Barahona and Mahjob fond a defining system of the stable set polytope for a graph with a ct-set of cardinality 2. We extend this reslt to ct-sets composed of a complete graph mins an edge and se the new theorem to derive a class of facets. 1 Preliminaries and Introdction Let G = (V, E) be a simple ndirected graph. A set S V of pairwise nonadjacent vertices defines a stable set. Let S := {S V S is a stable set}. Each S S has an incidence vector x S R V, where x S (v) = 1 if v S and x S (v) = 0 otherwise. Let P(G) be the convex hll of all x S sch that S S. P(G) is a fll-dimensional polytope and has a niqe (p to positive scalar mltiples) minimal defining system. Let V V and E(V ) := {v E, v V }. The sbgraph indced by V is G[V ] := (V, E(V )). For v V, define N G (v) := { V v E} and N G (v) := N G (v) {v}. A complete graph consists of pairwise adjacent vertices. A vertex set K V is complete whenever G[K] is a complete sbgraph. A maximal complete sbgraph defines a cliqe. A vertex v V is simplicial if G[N(v)] is complete. A ct-set C V decomposes G into a pair of proper sbgraphs (G 1, G 2 ) sch that C = V (G 1 ) V (G 2 ) and all paths from G 1 to G 2 intersect C. Chvátal [2] showed that the nion of defining systems of P(G 1 ) and P(G 2 ) defines P(G) when G[C] is a cliqe. Barahona and Mahjob [1] defined P(G) based on systems related to P(G 1 ) and P(G 2 ) when C 2. We extend this reslt to the case where G[C] is a complete graph mins an edge. Section 2 contains reslts necessary to extend Barahona and Mahjob s theorem. Section 3 generalizes their theorem. Section 3 refers to reslts from [1]. Section 4 applies the new theorem to derive a class of facets for the stable set polytope called diamonds. Section 5 smmarizes the reslts and discsses ftre research. bjm4@rice.ed, ivhicks@.rice.ed Department of Comptational and Applied Mathematics Rice University Hoston, 77005, USA. 1
v w 1 G k C\{,v} w 3 w 2 2 Spport Graphs Figre 1: The agmented graph G k. Sppose G has a ct-set C consisting of a nonadjacent pair of vertices. To obtain a defining system for P(G), Barahona and Mahjob [1] attach to C a new set of vertices {w i }. This agmentation defines a graph G. P( G) has a facet which projects along the sbspace of {w i } variables to define P(G). We generalize this method to the case where G[C] is a complete graph mins an edge. Section 3 analyzes the decomposition of G into the pair ( G 1, G 2 ). Here, we determine how the spport graphs of facets for P( G k ) interact with the {w i } vertices. Let a T x b be a nontrivial facet of P(G). Nontriviality implies b > 0 and a v 0 for all v V. In this section, all facets are assmed to be nontrivial. Define the following sets: V a := {v V a v > 0} and F a := {S S a T x S = b}. The spport graph of a T x b is defined as G a := G[V a ], the sbgraph indced by V a. Remark 1. Given a facet a T x b, F a consists of maximal stable sets in G a. In Section 3, we partition ineqalities based on their intersection with the set {w i }. Lemma 1 redces the nmber of partition sets. Recall that since P(G) is fll-dimensional, the sets S F a collectively satisfy no eqations other than scalar mltiples of a T x = b. Lemma 1. If a T x b is a non-cliqe facet, then G a contains no simplicial vertex. Proof. Sppose v V a is simplicial in G a. Then K := N Ga (v) is a cliqe and there exists an S F a sch that S K =. Otherwise, v K xs (v) = 1 for all S F a, a contradiction becase a T x b is not a cliqe ineqality. Observe that S is not a maximal stable set in G a, since S {v} is a feasible stable set. This contradicts Remark 1. Sppose G = (G 1, G 2 ) has a ct-set C where G[C] is a complete graph mins an edge. Notice G[C] has a stable set {, v}. For k {1, 2}, add the {w i } vertices to G k sch that N Gk (w 1 ) = {w 2 } (C \ {}), N Gk (w 2 ) = {w 1, }, and N Gk (w 3 ) = C. See Figre 1 for the agmented graph G k. The heavy edges denote joins (see [4]). For example, the edge between and C \ {, v} indicates that is adjacent to every vertex in C \ {, v}. Lemma 2. Let, v C be nonadjacent and C := C {w 1, w 2, w 3 }. For k {1, 2}, F k := {x P( G k ) z C x(z) = 2} is a facet for P( G k ). Moreover, no other facet contains all the vertices of C in its spport. 2
Proof. We show that F k is a facet for P( G k [ C]) by bilding a fll-rank C C matrix whose colmns are incidence vectors of all stable sets which lie on F k. See Figre 2. The first three rows correspond to w 1, w 2, and w 3. The last rows correspond to and C \ {}, respectively. Let 1 v be the ( C -1)-dimensional colmn vector with a 1 in row v and 0 s elsewhere. We now lift the ineqality z C x(z) 2 to a facet of P( G k ). Since all maximal stable sets J in G k satisfy J C = 2, the lifting coefficients for vertices in V ( G k ) \ C are zero. Ths, the ineqality is a facet of P( G k ). Sppose another facet a T x b contains all vertices of C in its spport. By Remark 1, z C xs (z) = 2 for all S F a. It follows that F k coincides with the face indced by a T x b. 1 0 T 1 0 0 0 1 T 0 1 0 0 0 T 1 1 0 1 0 T 0 0 1 0 I ( C 1) ( C 1) 0 0 1 v Figre 2: The matrix A. Given a defining system for a polytope, the process of projecting along a sbspace of variables, say w 1 and w 2, is less complicated if the coefficients of w 1 and w 2 are binary. The following lemma allows the defining systems encontered in Section 3 to be pt in this form. Lemma 3 (Mahjob [3]). Given a facet a T x b, let w 1, w 2 V a be adjacent vertices in G a. If w 1 is simplicial in G a w 2 and w 2 is simplicial in G a w 1, then a w1 = a w2. Lemma 3 implies that a w1 = a w2 in any nontrivial facet containing both w 1 and w 2 in its spport. As a reslt, scaling these ineqalities by (1/a w1 ) = (1/a w2 ) will prodce ineqalities where both variables have binary coefficients. 3 Composition of Stable Set Polyhedra This section offers a straightforward extension of techniqes developed by Barahona and Mahjob. We will refer to reslts from [1]. Let G = (G 1, G 2 ) have a ct-set C where G[C] is a complete graph mins an edge. Constrct the agmented graph G by adding a new set of vertices {w i } to C, as in Section 2. Define C := C {w i }. P( G) has a facet F = {x P( G) z C x(z) = 2} sch that P(G) = proj w1,w 2,w 3 {F } = {x R G w R 3 s.t. (x, w) F }. The set C decomposes G into the pair ( G 1, G 2 ). In Section 2, it was shown that P( G k ) has a facet F k for k {1, 2}. 3
Lemma 4 (Barahona and Mahjob [1]). The facet F is defined by the nion of the systems that define F 1 and F 2. Lemma 4 relies on the existence of a fll-rank, sqare matrix of all incidence vectors for stable sets on F, F 1, and F 2. The matrix A constrcted in the proof of Lemma 2 (see Figre 2) implies that this lemma holds for the class of ct-sets C we are analyzing. In order to find a defining system for F, consider the defining system for P( G k ) (other than cliqe ineqalities involving the {w i } variables). Recall from Section 2 that the spport of a T x b is denoted by V a. Lemma 1 and Lemma 2 imply that the facet-defining ineqalities can be partitioned into three sets I k 1, I k 2,I k 3 defined as follows: I k 1 := {a T i x b i V ai {w 1, w 2, w 3 } = } I k 2 := {at i x b i V ai {w 1, w 2, w 3 } = {w 1, w 2 }} I k 3 := {at i x b i V ai {w 1, w 2, w 3 } = {w 3 }}. Let V k = V (G k ). Lemma 3 and Lemma 4 imply that the defining system of F can be written as follows, k {1, 2}: 1. j V k a k ij x(j) bk i, for all i Ik 1 2. j V k a k ij x(j) + x(w 1) + x(w 2 ) b k i, for all i Ik 2 3. j V k a k ij x(j) + x(w 3) b k i, for all i Ik 3 4. j C\ x(j) + x(w 1) 1 5. j C\ x(j) + x(w 3) 1 6. j C\v x(j) + x(w 3) 1 7. x() + x(w 2 ) 1 8. x(w 1 ) + x(w 2 ) 1 9. j C x(j) = 2 10. x(j) 0, for all j Ṽk. The projection of this system along the sbspace of the {w i } variables is the polytope P(G). To define P(G), we proceed exactly as in [1]. Theorem 1. The polytope P(G) is defined by the nion of defining systems for P(G 1 ) and P(G 2 ), the non-negativity constraints, and the following facet-defining mixed ineqalities: a k ij x(j) + a l rj x(j) x(j) b k i + bl r 2 for k = 1, 2; l = 1, 2; k l; i Ik 2 ; r Il 3. j V k j V l j C Proof. See Theorem 3.5 and Corollary 3.7 in [1]. 4
4 Diamonds This section ses Theorem 1 to derive a class of facets for P(G). Let K 1,..., K 6 be sets of vertices sch that each K i is nonempty and complete. The graph G shown in Figre 3 is a member of a class of graphs which we call diamonds. The heavy edges denote joins. For example, an edge between K i and K j indicates that G[K i K j ] is complete. The size of the diamond is eqal to the nmber of sets K i. The diamond in Figre 3 has size 6, and z V x(z) 3 indces a facet for P(G). In general, facet-indcing diamonds have size 2n (where n > 1), a vertex sch that N G () = 2n i=1 K i, and a path P = p 1 p 2...p 2n 2 attached to the sets K 1,..., K 2n as shown in Figre 3. p 1 p 2 p 3 p 4 K 1 K 2 K 3 K 4 K 5 K 6 Figre 3: A diamond of size 6. Theorem 2. Let n > 1. If a diamond G has size 2n, then z V x(z) n indces a facet for P(G). Proof. The proof is by indction on n. Base case (n = 2): Choose v K 1 and w K 4. The diamond of size 4 has a 5-hole on the vertex set {p 1, p 2, w,, v}. Moreover, the odd-hole ineqality can be lifted to inclde all vertices in 4 i=1 K i. This implies that z V x(z) 2 indces a facet for P(G) as claimed. Indction step (n > 2): Sppose the theorem holds for all diamonds of even size less than 2n. The diamond of size 2n has a ct-set C = K 2n 3 {, p 2n 4 } which can be constrcted by removing an edge from a complete graph. Therefore, we apply Theorem 1. Figre 4 shows sbgraphs of the pair ( G 1, G 2 ). Let V 1 = V ( G 1 ) \ {w 1, w 2 } and V V ( G 2 ) \ {w 3 }. The graph on the left is a diamond of size 2n 2. By indction, 2 = x(z) n 1 (1) z V 1 is a facet for P( G 1 ). G2 has an odd-hole ineqality which lifts to obtain that x(z) 3 (2) z V 2 is a facet for P( G 2 ). Notice that ineqality (1) I3 1 and ineqality (2) I2 2. Theorem 1 gives the following facet-defining mixed ineqality for P(G) : x(z) + x(z) n 1 + 3 2 z V (G 1 ) z V (G 2 )x(z) z C 5
p 1... p 2n-4 p 2n-4 p 2n-3 p 2n-2 K 1 K 2... K 2n-3 w 3 w 1 K 2n-3 K 2n-2 K 2n-1 K 2n w 2 Figre 4: Sbgraphs of G 1 and G 2. Upon simplifying, we obtain that z V x(z) n is a facet for P(G) as claimed. Theorem 2 fails when the diamond has size that is odd and at least three. To see this, let G be the diamond of size 3 shown in Figre 5. G is perfect and not a cliqe, so G is not a spport graph for any facet of P(G). Now let G be the diamond of size 5 also shown in Figre 5. If G is the spport graph of a facet, then there mst exist 4 + 5 i=1 K i affinely independent maximal stable sets satisfying some eqation. However, no sch set exists. It follows by indction that a diamond of odd size is not a spport graph for any facet of P(G). 5 Conclsions and Ftre Work This paper generalized a theorem of Barahona and Mahjob concerning the composition of stable set polyhedra. The new reslt was applied to derive a class of facets called diamonds. Ftre research incldes sing composition to derive other classes of facets. An extension of Theorem 1 to more general ct-sets wold also be beneficial since composition can be applied recrsively. In other words, G can be decomposed into sbgraphs G 1,..., G m sch that the defining system for each P(G i ) is known. For example, decompose G into a set of perfect graphs. The defining systems for each P(G i ) can then be composed to define P(G). Another idea is to constrct P(G) starting from the leaves of a tree or branch decomposition. These approaches have the potential to characterize the stable set polytope for graphs which admit a strctred decomposition, bt they reqire a more general form of Theorem 1. Generalizing Theorem 1 might reqire techniqes different from the lift and project method of Barahona and Mahjob. Finding a G k and F k with the correct strctre ap- p p 1 p 2 p 3 K 1 K 2 K 3 K 4 K 5 Figre 5: Diamonds with odd size. 6
pears to be difficlt. A sbtle reqirement is that G k [ C] has to have exactly C affinely independent maximm stable sets. Otherwise, the matrix A is not invertible and Lemma 4 fails. Withot this restriction, Theorem 1 wold have held for any ct-set which partitions into two cliqes. There exist many graphs G k [ C] with exactly C affinely independent maximm stable sets, bt the ineqalities which define F k mst also involve the w i vertices in a strctred way. This strctre wold most likely involve extending the reslts of Section 2 to prevent the projection step from becoming too complicated. 6 Acknowledgements The athors wold like to thank the referee for comments which helped improve the presentation of this paper. This research was partially spported by NSF grants DMI-0521209 and DMS-0729251. References [1] F. Barahona and A.R. Mahjob, Compositions of Graphs and Polyhedra II: Stable Sets, SIAM J. Discrete Math., 7 (1994), pp. 359-371. [2] V. Chvátal, On Certain Polytopes Associated with Graphs, Jornal of Combinatorial Theory (B), 18 (1975), pp. 138-154. [3] A. R. Mahjob, On the Stable Set Polytope of a Series-Parallel Graph, Mathematical Programming, 40 (1988), pp. 53-57. [4] D. West, Introdction to Graph Theory, Prentice Hall, New York, 1996. 7