2-clique-bond of stable set polyhedra

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1 2-clique-bond of stable set polyhedra Anna Galluccio, Claudio Gentile, and Paolo Ventura IASI-CNR, viale Manzoni 30, Rome (Italy) E mail: {galluccio,gentile,ventura}@iasi.cnr.it Abstract The 2-bond is a generalization of the 2-join where the subsets of nodes that are connected on each shore of the partition are not necessarily disjoint. If all the subsets are cliques we say that the 2-bond is a 2-clique-bond. We consider a graph G obtained as the 2-clique-bond of two graphs G 1 and G 2 and we study the polyhedral properties of the stable set polytope associated with this graph. In particular, we prove that a linear description of the stable set polytope of G is obtained by properly composing the linear inequalities describing the stable set polytopes of four graphs that are related to G 1 and G 2. We show how to apply the 2-clique-bond composition to provide the complete linear description of large classes of graphs. Keywords: Stable set polytope, graph compositions, 2-join, polyhedral combinatorics. 1 Introduction Given a graph G = (V,E) and a vector w Q V + of node weights, the stable set problem is to find a set of pairwise nonadjacent nodes (stable set) of maximum weight. The stable set polytope of G is the convex hull of the incidence vectors of the stable sets of G. This polytope has full dimension and it is usually denoted as STAB(G). A linear system Ax b,x 0 is said to be defining for STAB(G) if STAB(G) = {x R V : Ax b,x 0}. Finding the defining linear system for ST AB(G) allows us to formulate the original optimization problem as the linear program max{w T x : Ax b,x 0}. Since the stable set problem is NP-hard, it is unlikely to find such a system for general graphs. Nevertheless the facial structure of the stable set polytope has been one of the major topics in polyhedral combinatorics and results concerning the facets of ST AB(G) have been provided continuously since early 70 s [33, 38, 29, 23, 8, 32, 35]. Besides the description of new classes of facets, there have been widely investigated composition procedures that are able to build new families of facets for the stable set polytope starting from facets of lower dimensional polytopes. These compositions are usually based on graph operations: for instance, the sequential lifting defined by Padberg [33] is based on the extension of a graph with an additional node, the procedure defined 1

2 by Wolsey [42] is based on edge subdivision, the composition of Barahona and Mahjoub [2] is based on the 2-node cutset. In addition to those listed above, a number of graph compositions were introduced in the attempt to solve the Strong Perfect Graph Conjecture (now Perfect Graph Theorem [9]): clique substitution [3], graph substitution [26], join [4, 15], amalgam [7], 2-amalgam [14]. All these operations were proved to preserve perfectness but many of them also have interesting polyhedral counterparts. In particular the knowledge of polyhedral descriptions of the smaller systems yields a description for the composed system in the case of: graph substitution [13], clique cutset composition [13], join [16], amalgam [6], and further generalizations [31, 27, 28]. The graph composition we consider is the following: Let G 1 and G 2 be two graphs. Let a i 0 and bi 0 be two adjacent nodes of G i and let A i = N(a i 0 ) \ {bi 0 } and B i = N(b i 0 ) \ {ai 0 }, for i = 1,2. The 2-bond of G 1 and G 2 along the edges a 1 0 b1 0 and a2 0 b2 0 is the graph G obtained by deleting the nodes a i 0 and bi 0, for each i = 1,2, and joining every node in A 1 with every node in A 2 and every node of B 1 with every node of B 2. If the sets A i and B i, i = 1,2, are disjoint the 2-bond is actually the 2-join defined in [34]. The 2-bond generalizes the 2-amalgam defined by Cornuejols and Cunningham [14] as follows. For i = 1,2, let G i be a graph, a i 0,bi 0 V (G i) be two adjacent nodes, K i {a i 0,bi 0 } be a clique in G i such that N(K i ) = N(a 0 i ) N(b0 i ), and K 1 = K 2. The 2-amalgam of G 1 and G 2 is the graph G obtained by deleting nodes a i 0,bi 0, for i = 1,2, identifying K 1 with K 2, and joining every node in N(a 1 0 ) with every node in N(a 2 0 ) and every node in N(b1 0 ) with every node in N(b2 0 ). Each graph G obtained as 2-amalgam can also be obtained as 2-bond of the graph G 1 and the graph G 2 \ K 2. Note that the 2-bond does not require that N(K i ) = N(a 0 i ) N(b0 i ) and so it generalizes the 2-amalgam. If A i and B i, i = 1,2, are (not necessarily disjoint) cliques we speak of a 2- clique-bond. In this paper, we study the polyhedral properties of the 2-clique-bond. In particular, we describe the structure of the stable set polytope of a graph G obtained as the 2-clique-bond of two smaller graphs G 1 and G 2 by showing how to obtain a linear description of ST AB(G) provided that the linear descriptions of four lower dimensional polytopes related to G 1 and G 2 are known. We also prove that the 2-clique-bond preserves the properties of inequalities of being facet defining. In sections 3 and 4 we define an instrumental graph composition, named W 3 - composition, that allows us to identify which facet defining inequalities play a role in the 2-clique-bond of stable set polyhedra and how these inequalities have to be composed. In section 5, we prove the main result of the paper. We show that the stable set polytope of a graph G, obtained as the 2-clique-bond of G 1 and G 2 along the edges e 1 and e 2, has a linear description that depends only on the linear descriptions of STAB(G i ) and STAB(G i /e i ), i = 1,2, where G i /e i represents the graph G i after contracting the edge e i. Finally we explain how the 2-clique-bond composition can be performed iteratively and how it can be applied in different directions: to define new facet defining inequalities of STAB(G), to build new classes of graphs whose stable set polytope is easy to describe, or to find the defining linear system of the stable set polytope 2

3 of well-known families of graphs. Samples of these applications are illustrated in Section 5 and mainly in Section 6. 2 Preliminaries Let G = (V,E) be any finite, simple and connected graph with node set V and edge set E. An edge e E with endnodes u and v is denoted by uv. A graph G together with a node weighting w Q V + is denoted as (G,w). We denote by α(g,w) the maximum weight of a stable set of G and we refer to α(g) = α(g,1) (1 being the vector of all ones) as the stability number of G. We denote by δ(v) the star of v, i.e., the set of edges of E having v as an endnode and by N(v) the neighbourhood of v, i.e., the set of nodes of V adjacent to v. We also denote by G \ A the subgraph of G induced by V \ A where A V, and by G/e the graph obtained by contracting the edge e of G (the contraction of an edge e is performed by identifying the endnodes of e and by removing loops and multiple edges). A node of V is simplicial if its neighbourhood consists of a clique. An edge e = v 1 v 2 of E is simplicial if N(v 1 )\{v 2 } and N(v 2 )\{v 1 } are two nonempty cliques of G. A clique cutset of G is a complete subgraph whose removal disconnects G. Given a vector β R m and a subset S {1,...,m}, let β(s) = i S β i. Moreover, define x S R m as the incidence vector of S. A linear inequality j V β jx j β 0 is valid for STAB(G) if it holds for all x STAB(G). For short, we also denote a linear inequality β T x β 0 as (β,β 0 ). A valid inequality for STAB(G) defines a facet of STAB(G) if and only if it is satisfied as an equality by V affinely independent incidence vectors of stable sets of G (called roots or tight solutions). The facet defining inequalities for ST AB(G) constitute the unique nonredundant defining linear system of STAB(G). We say that a stable set S is tight for (β,β 0 ) if β(s) = β 0 and that S violates (β,β 0 ) if β(s) > β 0. We denote by S (G) the set of all stable sets of G and by S MAX (G,w) the set {S S (G) w(s) = α(g,w)}. Observe that if (γ,γ 0 ) is facet defining for STAB(G) then S MAX (G,γ) is the set of the roots of (γ,γ 0 ). The proofs in this paper use basic concepts of (integer) linear programming that we summarize in the following. Lemma 1 Let (γ,γ 0 ) be a facet defining inequality for STAB(G). Then, for every valid inequality (β,β 0 ) that, up to positive multiplications, is not (γ,γ 0 ), there exists a stable set S S MAX (G,γ) that is not tight for (β,β 0 ) (i.e. β(s) < β 0 ). Proof. Let (β,β 0 ) be one such inequality. If β T x S = β 0 for all S S MAX (G,γ), then (β,β 0 ) contains all the roots of (γ,γ 0 ). As a consequence, since STAB(G) is full dimensional, either (γ,γ 0 ) is a positive multiple of (β,β 0 ) or it is not a facet defining inequality. In both cases we have a contradiction. Observation 1 Let G = (V,E) be a graph. For any node weighting function w Q V +, there exists a facet defining inequality (β,β 0) of STAB(G) such that β(s) = β 0 for any stable set S S MAX (G,w), i.e., S MAX (G,w) S MAX (G,β). 3

4 A basic result of linear programming [30] states that a point x P is an optimal solution of the optimization problem defined by the objective function w T x over the polyhedron P if and only if w can be expressed as a conic combination of the left hand side of the linear inequalities that define P and that are satisfied as equalities by x. As a consequence the following observation holds: Observation 2 Let G = (V,E) be a graph, w Q V + a node weighting function, and A x b a system of p linear inequalities that are facet defining for STAB(G) and such that β(s) = β 0 for any (β,β 0 ) (A x b ) and any S S MAX (G,w). If there does not exist λ R p + such that λt A = w, then there exists (β,β 0 ) that is facet defining for STAB(G) and with the following properties: there is no λ R p + such that λ T A = β ; β (S) = β 0 for any S S MAX(G,w). Chvátal [13] proved the following fundamental result concerning the stable set polytope of graphs containing clique cutsets: Theorem 2 Let G 1 = (V 1,E 1 ) and G 2 = (V 2,E 2 ) be two graphs. Let G 1 G 2 = (V 1 V 2,E 1 E 2 ) and G 1 G 2 = (V 1 V 2,E 1 E 2 ). If G 1 G 2 is a complete graph, then the defining linear system of STAB(G 1 G 2 ) is given by the union of the defining linear systems of STAB(G 1 ) and STAB(G 2 ). This implies that a graph supporting a facet defining inequality cannot contain a clique cutset. 3 W 3 -graphs In this section we introduce a new graph structure that will be used as an intermediate step to prove our final result. Definition 1 Let H be a graph, let A and B be two cliques of V (H). Let W 3 be another graph that is a clique of size four with V (W 3 ) = {t A,t B,t AB,t }. The graph G obtained as follows: V (G) = V (H) V (W 3 ) E(G) = E(H) E(W 3 ) F 1 F 2,where F 1 = {uv u {t A,t AB },v A} and F 2 = {uv u {t B,t AB },v B}. is called a W 3 -graph and denoted by G = (H,A,B,W 3 ). We now study the stable set polytope of W 3 -graphs. The first consideration is an easy consequence of Theorem 2: Corollary 3 If G is a W 3 -graph and Dx d defines STAB(G\{t }), then {Dx d,x ta + x tab + x tb + x t 1,x t 0} defines STAB(G). Thus, when G is a W 3 -graph, the only graph supporting a non trivial facet defining inequality for STAB(G) that contains t is the clique W 3 itself. In [19] we proved the following: 4

5 Proposition 4 Let G be a graph and let (β,β 0 ) be a facet defining inequality of STAB(G) which is neither a clique inequality nor a nonnegativity inequality. If uv is a simplicial edge then β u = β v. This result has some interesting consequences on the coefficients of the nodes of W 3 in facet defining inequalities of the stable set polytope of W 3 -graphs. Indeed: Proposition 5 Let G be a W 3 -graph and let (β,β 0 ) be a facet defining inequality (β,β 0 ) of STAB(G) which is neither a clique inequality nor a nonnegativity inequality. If β ta or β tb is nonzero then β ta = β tb = β tab. Proof. By Corollary 3, the edge t A t B is simplicial in every graph supporting a facet defining inequality (β,β 0 ) of STAB(G) which is neither a clique inequality nor a nonnegativity inequality. Thus, by Proposition 4, t A and t B have the same coefficient, namely β ta = β tb. By Lemma 1, there exists a stable set S S MAX (G,β) such that S {A {t A,t AB }} =, since A {t A,t AB } is a clique. This implies that t B S otherwise S {t A } would contradict the fact that S S MAX (G,β). Then β tb = β ta β tab because S\{t B } {t AB } is a stable set of G. As (β,β 0 ) is facet defining for STAB(G) and it is not a nonnegativity inequality, for each node u there exists a stable set that is tight for (β,β 0 ) containing u. In particular, there exists S S MAX (G,β) such that t AB S; then β tab β tb = β ta. Hence β ta = β tb = β tab in any facet defining inequality (β,β 0 ) of STAB(G) where t A and t B appear with nonzero coefficient. Given a W 3 -graph G, we say that an inequality (β,β 0 ) is a k-inequality, k {0,1,2,3,4}, if it is facet defining for STAB(G) and it has k nonzero coefficients among {β t,β ta,β tb,β tab }. Due to Proposition 5 and Corollary 3, we can classify the set of inequalities that define the stable set polytope of a W 3 -graph G as follows: Observation 3 Let G = (H,A,B,W 3 ) be a W 3 -graph and let (β,β 0 ) be a facet defining inequality of STAB(G) that is not a nonnegativity inequality. Then (β,β 0 ) is one of the following: i) a 0-inequality, i.e., β ta = β tb = β tab = β t = 0, ii) a 1-inequality with β ta = β tb = β t = 0 and β tab 0, iii) the 2-inequality u A x u + x ta + x tab 1, iv) the 2-inequality u B x u + x tb + x tab 1, v) a 3-inequality with β ta = β tb = β tab 0, vi) the 4-inequality x ta + x tb + x tab + x t 1. Note that inequalities iii), iv), and vi) are clique inequalities. 5

6 4 Compositions of W 3 -graphs Here we introduce a composition operation between W 3 -graphs. Definition 2 Given two W 3 -graphs G i = (H i,a i,b i,w i 3 ) with i = 1,2, the W 3- composition of G 1 and G 2 produces a new graph G by deleting the nodes V (W i 3 ) = {t i A,ti B,ti AB,ti }, for each i = 1,2, and joining every node of A 1 with every node of A 2 and every node of B 1 with every node of B 2. In Fig. 1, we represent a W 3 -composition of G 1 and G 2 where A i B i = for i = 1,2. Notice that in general A i B i and the nodes in A i B i may be adjacent to any node in V (H i ), i = 1,2, respectively. H 2 G 1 A 2 B 2 A 2 B 2 G A 1 B 1 H 1 G 2 A 1 B 1 Figure 1: G is the W 3 -composition of G i = (H i,a i,b i,w3 i ) for i = 1,2. We now introduce a new class of inequalities for the stable set polytope of graphs obtained as W 3 -composition. Let β 1 x β0 1 be a 1-inequality that is valid for STAB(G 1 ) and let β 2 x β0 2 be a 3-inequality that is valid for STAB(G 2). By Observation 3, it is not difficult to see that the coefficients of β 1 x β0 1 and β 2 x β0 2 can be normalized so that β1 (t 1 AB ) = β2 (t 2 A ) = β2 (t 2 B ) = β2 (t 2 AB ) = 1. Then the composition of (β 1,β0 1) and (β2,β0 2 ) is the following inequality: βvx 1 v + βvx 2 v β0 1 + β0 2 1 (1) v V (H 1 ) v V (H 2 ) In the rest of the paper when composing inequalities (β 1,β0 1) and (β2,β0 2 ) we will always assume that the nonzero coefficients on W3 1 and W 3 2 equal 1. Lemma 6 Let G be the W 3 -composition of two W 3 -graphs G 1 = (H 1,A 1,B 1,W3 1) and G 2 = (H 2,A 2,B 2,W3 2), let (β1,β0 1) be a 1-inequality that is valid for STAB(G 1), and (β 2,β0 2) a 3-inequality that is valid for STAB(G 2). If (β,β 0 ) is the composition of (β 1,β0 1) and (β2,β0 2 ) then it is valid for STAB(G). 6

7 Proof. To reach a contradiction, let S be a maximal stable set of G that violates (β,β 0 ), i.e., β(s) = β 1 (S V (H 1 )) + β 2 (S V (H 2 )) > β0 1 + β Clearly S (A 1 A 2 B 1 B 2 ) 2. Suppose first that S A 2 =. As (β 1,β0 1) is valid for STAB(G 1) and S V (H 1 ) is a stable set of G 1, β 1 (S V (H 1 )) β0 1 and, consequently, β2 (S V (H 2 )) > β But then (S V (H 2 )) t 2 A is a stable set of G 2 that violates (β 2,β0 2 ), a contradiction. So S A 2 and symmetric arguments prove also that S B 2. As a consequence, S A 1 = S B 1 =. Since (β 2,β0 2) is valid for STAB(G 2) and S V (H 2 ) is a stable set of G 2, we have that β 2 (S V (H 2 )) β0 2 and then β 1 (S V (H 1 )) > β But then (S V (H 1)) {t 1 AB } would be a stable set of G 1 violating (β 1,β0 1 ), a contradiction. As the W 3 -composition is symmetric, a result analogous to Lemma 6 holds when (β,β 0 ) is the composition of a 3-inequality for STAB(G 1 ) and a 1-inequality for STAB(G 2 ). In the remainder of this section we show how to obtain a linear description of STAB(G) starting from the linear description of the stable set polytopes of the two W 3 -graphs G 1 and G 2. Definition 3 Let G be the W 3 -composition of two W 3 -graphs G i = (H i,a i,b i,w3 i), i = 1,2. For any nonnegative node weighting function w for G, we define the generating weighting functions w i on V (G i ) for i = 1,2 as follows: w(u), if u V (H i ) α(h j \ B j,w) if u = t i w i A (u) = α(h j \ A j,w), if u = t i B α(h j \ (A j B j ),w), if u = t i α(h j,w), if u = t i AB with i,j = 1,2 and j i. We also say that the weighted graph (G,w) is generated by the weighted graphs (G 1,w 1 ) and (G 2,w 2 ). It is not difficult to observe that w i (t i ) wi (t i A ),wi (t i B ) wi (t i AB ) for i = 1,2 and that α(g,w) = α(g 1,w 1 ) = α(g 2,w 2 ). Definition 4 Let w be a node weighting function for G and let w i be the corresponding generating weighting functions on V (G i ), i = 1,2. Two stable sets S 1 S MAX (G 1,w 1 ) and S 2 S MAX (G 2,w 2 ) are said to generate S S MAX (G,w) if S = S 1 S 2 \ (W 1 3 W 2 3 ). Notice that for each S S MAX (G,w) we can produce two stable sets S 1 S MAX (G 1,w 1 ) and S 2 S MAX (G 2,w 2 ) that generate S in the following way: S i V (H i ) = S V (H i ) and t i A, if S A j,s B j = S i {t i A,ti B,ti AB,ti } = t i B, if S A j =,S B j t i AB, if S A j,s B j t i, if S A j =,S B j = with i,j = 1,2 and j i. 7

8 If S 1 S MAX (G 1,w 1 ) and S 2 S MAX (G 2,w 2 ) generate S S MAX (G,w) then the following conditions hold: w(s) = α(g,w) = w(s 1 V (H 1 )) + w(s 2 V (H 2 )) = = w 1 (S 1 ) = α(g 1,w 1 ) = w(s 1 V (H 1 )) + w 1 (S 1 V (W 1 3 )) = = w 2 (S 2 ) = α(g 2,w 2 ) = w(s 2 V (H 2 )) + w 2 (S 2 V (W 2 3 )) = = w 1 (S 1 V (W 1 3 )) + w2 (S 2 V (W 2 3 )). It is easy to prove the following: Observation 4 If S i S MAX (G i,w i ), then there always exists S j S MAX (G j,w j ) such that S i and S j generate a stable set S S MAX (G,w), for i,j {1,2} and j i. Moreover, the following property holds: Lemma 7 Let (G,w) be generated by (G 1,w 1 ) and (G 2,w 2 ), and let S i S MAX (G i,w i ) and S j S MAX (G j,w j ) generate S S MAX (G,w). If t i AB S i and u S j, with u {t j A,tj B,tj AB }, i,j {1,2} and j i, then wj (u) = w j (t j ). Proof. We first consider u = t j A. Let S = S i \ {t i AB } and then assume, by contradiction, that w j (t j A ) > wj (t j ). It follows that there exists a stable set S in H i \ B i such that w i (S ) > w j (t j ) = α(h i \ (A i B i ),w) = w i (S ). But then (S \ S ) S would be a stable set of G whose weight is greater than α(g,w), contradicting the fact that S S MAX (G,w). Similar arguments prove the cases u = t j B and u = tj AB. Theorem 8 Let G be the W 3 -composition of two W 3 -graphs G 1 = (H 1,A 1,B 1,W3 1) and G 2 = (H 2,A 2,B 2,W3 2 ). Then STAB(G) is described by the following inequalities: nonnegativity inequalities; 0-inequalities of STAB(G 1 ) or STAB(G 2 ); clique inequalities induced by A 1 A 2 and B 1 B 2 ; inequalities that are composition of: either a 3-inequality that is facet defining for STAB(G 1 ) and a 1-inequality that is facet defining for STAB(G 2 ), or a 1-inequality that is facet defining for STAB(G 1 ) and a 3-inequality that is facet defining for STAB(G 2 ). Proof. Suppose by contradiction that there exist a graph G and an inequality (γ,γ 0 ) that is facet defining for STAB(G) and that is not a positive multiple of any inequality listed in the thesis. Let G be such a graph with the minimum number of nodes. Then we may assume that (γ,γ 0 ) is fully supported by G. Now consider the vector γ as a node weighting of G and let γ i be the generating weighting functions γ i of γ on V (G i ), i = 1,2, as in Definition 3. Because of Observation 1, there exist two inequalities (β 1,β 1 0 ) and (β2,β 2 0 ) such that (β i,β i 0 ) is facet defining for STAB(G i) and such that all S i S MAX (G i,γ i ) 8

9 are tight for (β i,β0 i), i = 1,2. Since G i are W 3 -graphs, each non trivial inequality that is facet defining for STAB(G i ), i = 1,2, is of type i),...,vi) by Observation 3. Furthermore, as (γ,γ 0 ) is fully supported by G, there is at least one node u i V (H i ) with γ i (u i ) > 0 for i = 1,2. By Observation 2, this implies that, if S i S MAX (G i,γ i ), then there exists at least one facet defining inequality for STAB(G i ) that is satisfied as an equality by S i and that has nonzero coefficient on the node u i. As a consequence, there exists a facet defining inequality for STAB(G i ) that has nonzero coefficient on the node u i and that is satisfied to equality by all S i S MAX (G i,γ i ). Hence, we can assume that both (β 1,β0 1) and (β2,β0 2 ) are restricted to be in the set i),...,v). We first consider the case when (β i,β0 i ) is a 0-inequality for some i {1,2}. Without loss of generality let i = 1, i.e., β 1 (W3 1) = 0. Then, let S S MAX(G,γ), and let S 1 S MAX (G 1,γ 1 ) and S 2 S MAX (G 2,γ 2 ) generate S: as S V (H 1 ) = S 1 V (H 1 ) and, by assumption, S 1 is tight for (β 1,β0 1), then also S satisfies (β1,β0 1) as an equality. As this holds for any S S MAX (G,γ) and (γ,γ 0 ) is not a positive multiple of (β 1,β0 1 ), we get a contradiction of Lemma 1. We now suppose that (β i,β0 i ) is the 2-inequality of type iii) for some i {1,2} and without loss of generality let i = 1. Let S S MAX (G,γ) with S (A 1 A 2 ) = (such a stable set exists because of Lemma 1 applied to the clique inequality induced by the nodes of A 1 A 2 ) and let S 1 S MAX (G 1,γ 1 ) and S 2 S MAX (G 2,γ 2 ) generate S. As S 1 S MAX (G 1,γ 1 ) and (β 1,β0 1) is of type iii), then S 1 (A 1 {t 1 A,t1 AB }), i.e. there exists u {t1 A,t1 AB } contained in S 1. As S (A 1 A 2 ) =, γ(s 2 V (H 2 )) = γ 1 (t 1 B ). If u = t 1 AB, then S 1 (A 1 B 1 ) =. Moreover, the stable set S = S 1 \ {t 1 AB } {t 1 B } is not tight for (β1,β0 1) and, as by definition every stable set in S MAX(G 1,γ 1 ) is tight for (β 1,β0 1), it follows that S does not belong belong to S MAX (G 1,γ 1 ). As a consequence, γ 1 (t 1 AB ) > γ1 (t 1 B ), and so there exists a stable set S of H 2 such that γ(s ) > α(h 2 \ A 2,γ) = γ 1 (t 1 B ) = γ(s 2 V (H 2 )). But then S = (S V (H 1 )) S would be a stable set (because S (A 1 B 1 ) = ) with γ( S) > γ(s), thus violating the fact that S S MAX (G,γ). If u = t 1 A then we may assume that S 1 B 1 since otherwise S 1 \{t 1 A } {t1 AB } S MAX (G 1,γ 1 ) and we are in the previous case. Hence, γ(s 2 V (H 2 )) = γ 1 (t 1 ) and we can use arguments similar to the above ones to prove that γ 1 (t 1 A ) > γ1 (t 1 ). This implies that there exists a stable set S of H 2 \ B 2 such that γ(s ) > α(h 2 \ (A 2 B 2 ),γ) = γ 1 (t 1 ). As S = (S 1 V (H 1 )) S is a stable set (because S 1 A 1 = ) with γ( S) > γ(s), it would violate again the fact that S S MAX (G,γ). It follows that (β i,β0 i ) is not an inequality of type iii) for i = 1,2 and, symmetrically, it is not a 2-inequality of type iv) for i = 1,2. It remains to consider the following non symmetric cases: a) (β 1,β0 1) and (β2,β0 2 ) are both 1-inequalities Let S S MAX (G,γ) be such that S (A 1 A 2 ) = (such a stable set exists by Lemma 1). Observe that S (B 1 B 2 ) 1 and assume, without loss of generality, S B 1 =. Let S 1 S MAX (G 1,γ 1 ) and S 2 S MAX (G 2,γ 2 ) be two stable sets generating S. In particular, as S 1 S MAX (G 1,γ 1 ), S 1 is tight for the 1-inequality (β 1,β0 1) and it is not difficult to see that t1 AB S 1. Indeed, if t 1 AB / S 1, then S 1 \ {t 1 A,t1 B,t1 } {t1 AB } violates (β1,β0 1 ), a contradiction. 9

10 Moreover, we have that γ 1 (u) < γ 1 (t 1 AB ), for u {t1 A,t1 B,t1 }. Indeed, if γ(u) = γ(t 1 AB ) for some u {t1 A,t1 B,t1 }, then S = S 1 \ {t 1 AB } {u} would be a maximum stable set in S MAX (G 1,γ 1 ) such that β 1 (S ) = β0 1 β1 (t 1 AB ) < β1 0, violating the hypothesis that all the stable sets in S MAX (G 1,γ 1 ) are tight for (β 1,β0 1). Hence γ 1 (u) < γ 1 (t 1 AB ) for u {t1 A,t1 B,t1 }. This implies that there exists a stable set S V (H 2 ) such that γ(s ) = α(h 2,γ) = γ 1 (t 1 AB ) > γ1 (t 1 B ) = α(h 2 \ A 2,γ) = γ(s V (H 2 )). But then, as S A 1 = and S B 1 =, S = (S V (H 1 )) S would be a stable set of G with γ( S) > γ(s), violating the fact that S S MAX (G,γ). A contradiction. b) (β 1,β0 1) and (β2,β0 2 ) are both 3-inequalities As for Case a), let S S MAX (G,γ) be such that S (A 1 A 2 ) =. Since S (B 1 B 2 ) 1 we can assume without loss of generality that S B 1 =. Let S 1 S MAX (G 1,γ 1 ) and S 2 S MAX (G 2,γ 2 ) be two stable sets generating S. As γ 1 (t 1 AB ) γ1 (u) for any u {t 1 A,t1 B,t1 }, there exists a stable set S 1 S MAX (G 1,γ 1 ) with t 1 AB S 1 and S 1 V (H 1 ) = S 1 V (H 1 ). Moreover, as γ 2 (t 2 AB ) γ2 (t 2 A ),γ2 (t 2 B ) γ2 (t 2 ), there exists a stable set S 2 S MAX (G 2,γ 2 ) containing v {t 2 A,t2 AB } and such that S 2 V (H 2 ) = S 2 V (H 2 ). Now observe that S 1 and S 2 do generate S. Furthermore, as S 2 is tight for (β 2,β0 2) and S 2 \ {v} {t 1 } is not tight for (β2,β0 2 ), then also γ 2 (v) > γ 2 (t 2 ), contradicting Lemma 7. c) (β 1,β0 1) is a 1-inequality and (β2,β0 2 ) is a 3-inequality Let (β,β 0 ) be the inequality (1) obtained by composing (β 1,β0 1) and (β2,β0 2). Here, we show that (β,β 0 ) is satisfied as an equality by each S S MAX (G,γ), so contradicting Lemma 1. Suppose on the contrary that there exists a stable set S S MAX (G,γ) such that β(s V (H 1 ))+β(s V (H 2 )) < β 0 = β0 1+β As the coefficients of (β 1,β0 1) and (β2,β0 2) are normalized so that β1 (t 1 AB ) = 1 and β 2 (t 2 AB ) = β2 (t 2 A ) = β2 (t 2 B ) = 1, this implies that S is not tight for (β,β 0 ) if and only if t 1 AB S 1 and u S 2, where u {t 2 A,t2 B,t2 AB } and, as usual, S 1 S MAX (G 1,γ 1 ) and S 2 S MAX (G 2,γ 2 ) generate S. As by hypothesis every stable set in S MAX (G 2,γ 2 ) is tight for (β 2,β0 2 ), it follows that γ 2 (u) > γ 2 (t 2 ), so contradicting Lemma 7. The next theorem shows that the description of ST AB(G) given by Theorem 8 is also minimal. Theorem 9 Let (β,β 0 ) be a composition of a facet defining 1-inequality (β 1,β 1 0 ) of STAB(G 1 ) and a facet defining 3-inequality (β 2,β 2 0 ) of STAB(G 2). Then (β,β 0 ) is facet defining for STAB(G). Proof. Suppose by contradiction that (β,β 0 ) is not facet defining for STAB(G); then there exists an inequality (γ,γ 0 ) that defines a facet of STAB(G) containing all the roots of (β,β 0 ). By Theorem 8, (γ,γ 0 ) is one of the following: nonnegativity inequalities; 10

11 0-inequalities of STAB(G 1 ) or STAB(G 2 ); clique inequalities induced by A 1 A 2 or B 1 B 2 ; inequalities that are composition of: either a 3-inequality that is facet defining for STAB(G 1 ) and a 1-inequality that is facet defining for STAB(G 2 ), or a 1-inequality that is facet defining for STAB(G 1 ) and a 3-inequality that is facet defining for STAB(G 2 ). First consider the case where (γ,γ 0 ) is a nonnegativity inequality or a 0-inequality and, without loss of generality, assume that it is defined over the nodes of G 1. As a consequence, its restriction (γ 1,γ0 1) over V (G 1) is facet defining for STAB(G 1 ) and then, by Lemma 1, there exists S 1 S MAX (G 1,β 1 ) such that S 1 is not tight for (γ 1,γ0 1). By Observation 4 there exists also S 2 S MAX (G 2,β 2 ) such that S 1 and S 2 generate S S MAX (G,β). By construction, such S is not tight for (γ,γ 0 ), contradicting the assumption. Suppose now that (γ,γ 0 ) is a clique inequality induced by A 1 A 2. Then again, as (β 1,β0 1) is facet defining for STAB(G 1), there exists S 1 S MAX (G 1,β 1 ) such that S 1 is not tight for the clique inequality defined by A 1 {t 1 A,t1 AB }; as a consequence S 1 {t 1 B,t1 }. By Observation 4, there exists S 2 S MAX (G 2,β 2 ) such that S 1 and S 2 generate S S MAX (G,β). As t 1 B S 1 or t 1 S 1 then we can choose S 2 such that S 2 A 2 =. As a consequence, S is not tight for (γ,γ 0 ), a contradiction. Consider now the case when (γ,γ 0 ) is a composition of a 1-inequality (γ 1,γ0 1) that is facet defining for STAB(G 1 ) and a 3-inequality (γ 2,γ0 2 ) that is facet defining for STAB(G 2 ). If (γ 1,γ0 1) is equivalent to (β1,β0 1) and (γ2,γ0 2 ) is equivalent to (β 2,β0 2), then (γ,γ 0) is equivalent to (β,β 0 ) and we are done. Then, without loss of generality, assume that (γ 1,γ0 1 ) cannot be obtained as a positive multiple of (β 1,β0 1). Hence, there exists S 1 S MAX (G 1,β 1 ) such that γ 1 (S 1 ) < γ0 1. Again, by Observation 4 there exists S 2 S MAX (G 2,β 2 ) such that S 1 and S 2 generate S S MAX (G,β). As γ 2 (S 2 ) γ0 2, then it is not difficult to see that, in order to have S tight for (γ,γ 0 ), it should be γ 1 (S 1 W3 1) + γ2 (S 2 W3 2 ) < 1. However, the validity of the composition of (γ 1,γ0 1) and (γ2,γ0 2 ) established by Lemma 6 is equivalent to the condition γ 1 ( S 1 W3 1) + γ2 ( S 2 W3 2) 1 for any pair S 1, S2 generating a maximal stable set S of G. So we get a contradiction. Analogous arguments apply to prove the case where (γ,γ 0 ) is a composition of a 3-inequality (γ 1,γ0 1) that is facet defining for STAB(G 1) and a 1-inequality (γ 2,γ0 2) that is facet defining STAB(G 2 ). 5 The main result In this section we prove the main result of the paper. We start by formally defining the 2-clique-bond. Definition 5 Let G 1 and G 2 be two graphs. Let a i 0 and bi 0 be two adjacent nodes of G i such that A i = N(a i 0 ) \ {bi 0 } and B i = N(b i 0 ) \ {ai 0 } are two cliques of V (G i), i = 1,2. The 2-clique-bond of G 1 and G 2 along the edges a 1 0 b1 0 and a2 0 b2 0 is the graph G obtained by deleting the nodes a i 0 and bi 0, for i = 1,2, and joining every node in A 1 with every node in A 2 and every node of B 1 with every node of B 2. 11

12 H W 5 G Figure 2: Graph G is the 2-clique-bond composition of H and W 5. Fig. 2 represents the 2-clique-bond composition of a graph H and a 5-wheel W 5 along the two simplicial edges depicted in bold. In the following we see how to compose inequalities that are valid for the stable set polytopes of some graphs associated with G 1 and G 2 in order to obtain valid inequalities for the stable set polytope of a graph G that is the 2-clique-bond of G 1 and G 2. Definition 6 Let G i be a graph with a simplicial edge e i = a i 0 bi 0, i = 1,2 and let G be the 2-clique-bond of G 1 and G 2 along e 1 and e 2. Let G i /e i be the graph obtained by contracting the edge a i 0 bi 0 into a single node zi 0. An inequality (β,β 0 ) of STAB(G) is said to be an even-odd combination of inequalities of STAB(G i ) and STAB(G j /e j ) for i,j = 1,2 and i j if it has the following form: βvx 1 v + βvx 2 v β0 1 + β0 2 1, (2) v V (G 1 \{a 1 0,b1 0 }) v V (G 2 \{a 2 0,b2 0 }) where either β 1 x β0 1 is a valid inequality for STAB(G 1) different from x a x b 1 0 1, β 2 x β0 2 is a valid inequality for STAB(G 2/e 2 ) different from the clique inequalities supported by A 2 {z0 2} or B 2 {z0 2}, and β1 = β 1 = β 2 = 1, a 1 0 b 1 0 z0 2 or β 1 x β0 1 is a valid inequality for STAB(G 1/e 1 ) different from the clique inequalities supported by A 1 {z0 1} or B 1 {z0 1}, β2 x β0 2 is a valid inequality for STAB(G 2 ) different from x a x b 2 0 1, and β 1 = β 2 = β 2 = 1. z0 1 a 2 0 b 2 0 Note that the conditions β 1 = β 1 and β 2 = β 2 are not restrictive by Proposition 4. Therefore it is always possible to normalize (β 1,β0 1) and (β2,β0 2 ) so that a 1 0 b 1 0 a 2 0 b 2 0 = β 1 = β 2 = 1 or β 1 = β 2 = β 2 = 1. b 1 0 z0 2 z0 1 a 2 0 b 2 0 β 1 a 1 0 In Fig. 3 it is shown how to obtain a valid inequality for the stable set of the graph G in Fig. 2 as an even-odd combination of a 5-wheel inequality of STAB(W 5 ) and an inequality of STAB(H/e) (where e is the simplicial edge involved in the 2- clique-bond composition). We can now prove the main result of the paper. 12

13 3x + 2x + x 4 2x + x 2 3x + 2x + x + 4x 6 Figure 3: Even-odd combination of inequalities Theorem 10 Let G i be a graph with a simplicial edge e i = a i 0 bi 0, i = 1,2, and let G be the 2-clique-bond of G 1 and G 2. Then STAB(G) is described by the following inequalities: nonnegativity inequalities; clique inequalities induced by A 1 A 2 and B 1 B 2 ; facet defining inequalities of STAB(G i ) with zero coefficients on the endnodes of e i for each i = 1,2; even-odd combinations of facet defining inequalities of STAB(G i ) and STAB(G j /e j ) for each i,j = 1,2 and i j. Proof. Build two W 3 -graphs Γ i = (G i \ {a i 0,bi 0 },A i,b i,w3 i ), i = 1,2 as described in Definition 1. It is not difficult to see that the W 3 -composition of Γ 1 and Γ 2 is isomorphic to G, thus STAB(G) can be derived from Theorem 8 by observing that: The 0-inequalities of STAB(Γ i ) have the same supporting graph of facet defining inequalities of STAB(G i ) with zero coefficients on the endnodes of e i, for i = 1,2, and therefore they are equivalent. The 1-inequalities of STAB(Γ i ) have the same supporting graph of facet defining inequalities of STAB(G i /e i ) that have nonzero coefficient on z0 i and are different from the clique inequalities supported by A i {z0 i } or B i {z0 i }, for i = 1,2. The graphs Γ i \ {t i,ti AB } are isomorphic to G i where t i A and ti B correspond to a i 0 and bi 0, respectively. Thus every 3-inequality of STAB(Γ i) corresponds to a facet defining inequality of STAB(G i ) having nonzero coefficients on a i 0 and b i 0 where the additional node ti AB, i = 1,2, has been sequentially lifted [33]. This follows from Proposition 5. The composition of 1-inequalities and 3-inequalities as described in (1) produces the same inequalities obtained by the even-odd combination (2) described in Definition 6. 13

14 Thus the theorem follows and, by Theorem 9, the provided linear description is also minimal. By Theorem 10 it is not difficult to verify that the graph G in Fig. 2 supports a nontrivial facet defining inequality obtained as the even-odd combination of the 5-wheel inequality of STAB(W 5 ) and the facet defining inequality of STAB(H/e) defined by Giles and Trotter in [23] (see Fig. 3). 6 Iterated 2-clique-bond compositions The results obtained so far can be applied in different directions: either to define new facet defining inequalities of STAB(G), or to build new classes of graphs whose stable set polytope has an explicit linear description, or to find the defining linear system of the stable set polytope of classes of graphs that are known. An example for the first direction is given by the inequality presented in Figure 3. In this section we mention other two possible applications. Up to this point we have considered only graphs that are obtained by performing a single 2-clique-bond composition on a given pair of graphs G 1 and G 2. In the following we focus on graphs obtained by repeated applications of the 2-clique-bond composition. Definition 7 Given a graph H, let Γ H = {e 1,...,e k } denote a set of nonadjacent simplicial edges of H. Let T be a family of graphs and let T i be a graph in T with a simplicial edge f i, i = 1,...,k. A graph H k obtained by k applications of the 2-clique-bond composition to H is defined as follows: H 0 = H, H i is the 2-clique-bond of H i 1 and T i along e i and f i, i = 1,...,k. Given two families of graphs H and T, the family of graphs H i, i = 1,...,k obtained from a graph H H by repeated applications of 2-clique-bond compositions with graphs in T is denoted by G(H, T ). Note that in Definition 7 the 2-clique-bond compositions are performed only along simplicial edges of H i 1 that were also simplicial in the original graph H. Accordingly with Definition 7 we define a larger family of inequalities that contains the even-odd combinations generated by repeated applications of the 2-cliquebond composition. Definition 8 Let L be a family of valid inequalities for the stable set polytope. An inequality (γ,γ 0 ) L if and only if either (γ,γ 0 ) L, or (γ,γ 0 ) is an even-odd combination of two inequalities in L. We say that L is closed under even-odd combination if L = L. 14

15 A well-known family of inequalities that is closed under even-odd combination is the family R of rank inequalities. Consider now a set L of valid inequalities for STAB(G) together with the following polyhedron LSTAB(G) = {x R V + x satisfies L}. As a generalization of perfect graphs, L-perfect graphs are defined as those graphs having LSTAB(G) = STAB(G) [24]. A natural question is the following: which graphs belong to the class of L-perfect graphs? Different families L of inequalities have been considered in the literature together with the corresponding classes of L-perfect graphs. We mention some of them in a nonexhaustive list: edge plus odd-hole inequalities and t-perfect graphs [13]; clique plus odd-hole inequalities and h-perfect graphs; rank inequalities and rank-perfect graphs [39]. In the previous section we state that, if G is the 2-clique-bond of G 1 and G 2 along the simplicial edges e 1 and e 2, the defining linear system for STAB(G) can be easily provided once the defining linear systems of STAB(G i ) and STAB(G i /e i ), i = 1,2 are known. So, an immediate consequence of Theorem 10 is: Corollary 11 Let G be the 2-clique-bond of G 1 and G 2 along the simplicial edges e 1 and e 2. If G i and G i /e i, i = 1,2 are L-perfect then G is L -perfect. Moreover, if L is closed under even-odd combination then G is L-perfect. This result can be applied iteratively to obtain classes of L-perfect graphs. In the following we denote by H/F the graph obtained from H by contracting all the edges of F Γ H, where Γ H is a set of nonadjacent simplicial edges of H. Theorem 12 Let H be a graph, F = {e 1,e 2,...,e k } Γ H, and let H k be obtained from H by k applications of the 2-clique-bond composition with graphs T i T, i = 1,...,k (as explained in Definition 7). If H and H/F are L-perfect for any F F, and if each T i and T i /f i, i = 1,...,k, is L-perfect, then H k is L -perfect. Proof. The proof is by induction on the number k of 2-clique-bond compositions. If k = 1 then H 1 is L -perfect by Corollary 11. Let k > 1 and let H k be the 2-clique-bond of H k 1 and T k along e k and f k. By observing that (L ) = L and by using Corollary 11, in order to prove that H k is L -perfect it suffices to show that the graphs T k, T k /f k, H k 1, and H k 1 /e k are L -perfect. By hypotheses T k and T k /f k are L-perfect and H k 1 is L -perfect by inductive hypothesis. Hence, it remains to show that H k 1 /e k is L -perfect. As the edges {e 1,e 2,...,e k } are nonadjacent, H k 1 /e k can be obtained by k 1 repeated applications of the 2-clique-bond of T 1, T 2,..., T k 1 to the graph H/e k along the simplicial edges {e 1,...,e k 1 }. Since, by hypothesis, H/e k and (H/e k )/F are L-perfect for any F {e 1,...,e k 1 }, it follows by the inductive hypothesis that H k 1 /e k is L -perfect. Thus the theorem follows. As a consequence of Theorem 12 one has that, if L is closed under even-odd combinations, then H k is L-perfect. In the following we explain how to use the 15

16 iterated 2-clique-bond composition to build new classes of graphs whose stable set polytope is well understood. Consider the class P of (P 5,gem)-free graphs defined in [1] as those graphs without P 5 and gem as induced subgraphs (P 5 is an induced path with 5 nodes and gem is the graph consisting of a node adjacent to all nodes of an induced path with 4 vertices). The stable set problem on this class of graphs has been studied both from the algorithmic and the polyhedral point of view. In particular, there exists a linear time algorithm to solve the maximum weight stable set in (P 5,gem)- free graphs [5] and it is known the complete linear description of their stable set polytope [36]. Starting from (P 5,gem)-free graphs we can build a much larger class of graphs where it is possible to extend most of the polyhedral results proved for (P 5,gem)-free graphs. Consider a graph H P and denote by Γ H the set of nonadjacent simplicial edges e = ab E(H) such that: (*) no node in N(a) \ {b} is adjacent to a node in N(b) \ {a}. Under the assumption (*) it is not difficult to check that the contracted graph H/e is still (P 5,gem)-free. Nevertheless the graphs resulting from iterated applications of the 2-clique bond composition as explained in Definition 7 may contain P 5 s. So, the class G(P, P) is a nontrivial generalization of the class of (P 5,gem)- free graphs. The class P is proved to be rank-perfect in [36]. Thus by Theorem 12, the following result holds: Theorem 13 The classes G(P, P) and G(H, P), where H is the family of line graphs, are R-perfect. Notice that, besides line graphs and bipartite graphs, a few classes of graphs are known to be rank-perfect, such as antiwebs [40] or some special subclasses of quasiline graphs [41]. The 2-clique-bond composition allows to considerably expand the classes of graphs which are known to be rank-perfect. Finally, we want to mention another relevant application of Theorem 12 concerning the stable set polytope of claw-free graphs. Using the decomposition theorem of claw-free graphs of Chudnovsky and Seymour [11], it is possible to represent every claw-free, not quasi-line graph, with stability number at least four, and with no 1-join as a graph obtained by iteratively performing the 2-clique-bond composition along simplicial edges a i 0 bi 0, i = 1,2, having the following further property: (**) the neighbourhood of A i B i coincides with (A i B i ) {a 0 i,b0 i },i = 1,2. When condition (**) is satisfied the graph obtained as the 2-clique-bond composition along a i 0 bi 0, i = 1,2, admits a generalized 2-join as defined by Chudnovsky and Seymour in [12], where V 0 = (A 1 B 1 ) (A 2 B 2 ). Accordingly with the construction defined in [12], take a graph G 0 that is a disjoint union of cliques and take a partition of V (G 0 ) into stable sets X 1,...,X k such that each X i satisfies X i = 2. Now, connect each pair of nodes in X i with an edge e i ; it is not difficult to see that these graphs form a subclass H of line graphs where each edge e i is simplicial and satisfies condition (**). 16

17 A strip (G,a 0,b 0 ) consists of a claw-free graph G together with two designated simplicial nodes a 0,b 0 V (G 0 ) called the ends of the strip. In the construction of claw-free, not quasi-line graphs with stability number at least four and with no 1-join, the building blocks identified by Chudnovsky and Seymour are the strips belonging to five different families of claw-free graphs that are named Z 1, Z 2, Z 3, Z 4, and Z 5. (The definition of these families is quite involved and we refer the interested reader to [12] for details). Observe now that if we add an edge between the ends of a strip, that edge is simplicial. So, we can denote by Z i + the family of closed strips obtained by adding an edge between the ends of each strip (G,a 0,b 0 ) where G Z i, i = 1,...,5. If we set Z = {Z 1 +, Z+ 2, Z+ 3, Z+ 4, Z+ 5 }, then we can rephrase part of the global structure theorem of Chudnovsky and Seymour as follows: Theorem 14 [12] Every claw-free graph with stability number at least four, that is not quasi-line and admits no 1-join, belongs to the class G(H, Z ). Chudnovsky and Seymour [10] observed that the stable set polytope of graphs in G(H, Z 1 + ) are rank-perfect (see [37] for a proof). However, it is known that the rank inequalities do not suffice to describe the stable set polytope of claw-free graphs since the 5-wheel inequalities and the geared inequalities [18] need to be added. In [19] we defined the following family of inequalities that is closed under even-odd combinations provided that (**) holds for the simplicial edges involved in the iterated 2-clique-bond composition: G = {nonnegativity, rank, lifted 5-wheel, lifted multiple geared rank inequalities}. In [20, 21] we proved that all graphs in Z and their contraction along simplicial edges satisfying (**) are G-perfect. This result together with Theorem 12 and Theorem 14 yields the following: Theorem 15 Every claw-free, not quasi-line graph with stability number at least four and with no 1-join is G-perfect. In [25], Grötschel, Lovász and Schrijver asked for an explicit linear description of the stable set polytope of claw-free graphs. The linear description of STAB(G) is known for graphs with stability number two (see [35]) and for graphs containing a 1-join (see [13]). For the subclass of claw-free graphs consisting of quasi-line graphs (graphs such that the neighbourhood of each node can be partitioned into two cliques), a (not minimal) linear description of STAB(G) has been provided by Chudnovsky and Seymour [10] and Eisenbrand et al. [17]. Theorem 15 (and the characterization in [22]) yields a minimal linear description of the stable set polytope of claw-free, not quasi-line graphs with stability number at least four and leaves open this longstanding question only for the case α(g) = 3. References [1] A. Brandstädt and D. Kratsch. On the structure of (P 5,gem)-free graphs. Discrete Mathematics, 145: ,

18 [2] F. Barahona and A.R. Mahjoub. Compositions of graphs and polyhedra II: stable sets. SIAM J. on Discr. Math., 7: , [3] C. Berge. Graphs and Hypergraphs. North-Holland, Amsterdam, [4] R. E. Bixby. A composition for perfect graphs. Annals of Discr. Math., 21: , [5] H.L. Bodlaender, A. Brandstädt, D. Kratsch, M. Rao, and J. Spinrad. On algorithms for (P 5,gem)-free graphs. Theoret. Comput. Sci., 349:2 21, [6] E. Burlet and J. Fonlupt. Polyhedral consequence of the amalgam operation. Discrete Mathematics, 130:39 55, [7] M. Burlet and J. Fonlupt. Polynomial algorithm to recognize a Meyniel graph. Annals of Discr. Math., 21: , [8] E. Cheng and W. H. Cunningham. Wheel inequalities for stable set polytopes. Mathematical Programming, 77: , [9] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas. The Strong Perfect Graph Theorem. Annals of Math., 164:51 229, [10] M. Chudnovsky and P. Seymour. The structure of claw-free graphs. In Surveys in Combinatorics, volume 327 of London Math. Soc. Lecture Notes, pages Cambridge University Press, [11] M. Chudnovsky and P. Seymour. Claw-free graphs IV: Decomposition theorem. J. Comb. Th. B, 98(5): , [12] M. Chudnovsky and P. Seymour. Claw-free graphs V: Global structure. J. Comb. Th. B, 98: , [13] V. Chvátal. On certain polytopes associated with graphs. J. Comb. Th. B, 18: , [14] G. Cornuéjols and W. H. Cunningham. Compositions for perfect graphs. Discrete Mathematics, 55: , [15] W. H. Cunningham and J. Edmonds. A combinatorial decomposition theory. Canad. J. Math., 32: , [16] W.H. Cunningham. Polyhedra for composed independence systems. Annals of Discr. Math., 16:57 67, [17] F. Eisenbrand, G. Oriolo, G. Stauffer, and P. Ventura. The stable set polytope of quasi-line graphs. Combinatorica, 28(1):45 67, [18] A. Galluccio, C. Gentile, and P. Ventura. Gear composition and the stable set polytope. Operations Research Letters, 36: , [19] A. Galluccio, C. Gentile, and P. Ventura. Gear composition of stable set polytopes and G-perfection. Mathematics of Operations Research, 34: ,

19 [20] A. Galluccio, C. Gentile, and P. Ventura. The stable set polytope of claw-free graphs with large stability number I: Fuzzy antihat graphs are W-perfect. Technical Report 10-06, Istituto di Analisi dei Sistemi ed Informatica A. Ruberti - CNR, Submitted. [21] A. Galluccio, C. Gentile, and P. Ventura. The stable set polytope of claw-free graphs with large stability number II: Striped graphs are G-perfect. Technical Report 10-07, Istituto di Analisi dei Sistemi ed Informatica A. Ruberti - CNR, Submitted. [22] A. Galluccio and A. Sassano. The rank facets of the stable set polytope for claw-free graphs. J. Comb. Th. B, 69(1):1 38, [23] R. Giles and L.E. Trotter. On stable set polyhedra for K 1,3 -free graphs. J. Comb. Th. B, 31(3): , [24] M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2): , [25] M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization. Springer Verlag, Berlin, [26] L. Lovász. Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics, 2: , [27] F. Margot. Composition de polytopes combinatoires: une approche par projection. PhD thesis, École Polytechnique Fédéral de Lausanne, Switzerland, [28] B. McClosky and I. V. Hicks. Composition of stable set polyhedra. Operations Research Letters, 36: , [29] G. Nemhauser and L.E. Trotter Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6:48 61, [30] G.L. Nemhauser and L.A. Wolsey. Integer and Combinatorial Optimization. John Wiley & Sons, Inc., [31] P. Nobili and A. Sassano. Polyhedral properties of clutter amalgam. SIAM J. on Discr. Math., 6(1): , [32] G. Oriolo. Clique family inequalities for the stable set polytope for quasi-line graphs. Discrete Applied Mathematics, 132: , [33] M. W. Padberg. On the facial structure of vertex packing polytope. Mathematical Programming, 5: , [34] A. Schrijver. Combinatorial optimization. Springer Verlag, Berlin, [35] F.B. Shepherd. Near-perfect matrices. Mathematical Programming, 64: ,

20 [36] C. De Simone and R. Mosca. Stable set and clique polytopes of (P 5,gem)- graphs. Discrete Mathematics, 307: , [37] G. Stauffer. On the stable set polytope of claw-free graphs. PhD thesis, École Polytechnique Fédéral de Lausanne, Switzerland, [38] L.E. Trotter Jr. A class of facet producing graphs for vertex packing polyhedra. Discrete Mathematics, 12: , [39] A. Wagler. Rank-perfect and weakly rank-perfect graphs. Mathematical Methods of Operations Research, 95: , [40] A. Wagler. Antiwebs are rank-perfect. 4OR, 2: , [41] A. Wagler. On rank-perfect subclasses of near-bipartite graphs. 4OR, 3: , [42] L.A. Wolsey. Further facet generating procedures for vertex packing polytopes. Mathematical Programming, 11: ,