2 -clique-bond of stable set polyhedra

Discrete Applied Mathematics 161 (2013) 1988–2000

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2-clique-bond of stable set polyhedra Anna Galluccio, Claudio Gentile, Paolo Ventura ∗ Istituto di Analisi dei Sistemi ed Informatica ‘‘A. Ruberti’’, Consiglio Nazionale delle Ricerche (IASI-CNR), Viale Manzoni 30, 00185 Roma, Italy

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Article history: Received 19 May 2011 Received in revised form 15 February 2013 Accepted 18 February 2013 Available online 22 March 2013 Keywords: Stable set polytope Graph compositions 2-join Polyhedral combinatorics

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. The 2-clique-bond composition builds a graph G admitting a 2-clique-bond starting from two graphs G1 and G2 . 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 G1 , G2 and two other related graphs. We explain how to apply iteratively the 2-clique-bond composition to provide the complete linear description of the stable set polytope of new classes of graphs. © 2013 Elsevier B.V. All rights reserved.

1. Introduction |V |

Given a graph G = (V , E ) and a vector w ∈ Q+ of node weights, the stable set problem is the problem of finding 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 by STAB(G). A linear system Ax ≤ b, x ≥ 0 is said to be defining for STAB(G) if STAB(G) = {x ∈ RV : Ax ≤ b, x ≥ 0}. Finding a defining linear system for STAB(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 a polytime separable 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 STAB(G) have been provided continuously since early 1970s [30,33,26,21,8,29,32,34,17,18]. Besides the description of new classes of facets, several composition procedures have been investigated to build 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 [30] is based on the extension of a graph with an additional node, the procedure defined by Wolsey [37] is based on edge subdivision, and the composition of Barahona and Mahjoub [1] 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 Strong Perfect Graph Theorem [9]): clique substitution [2], graph substitution [23], join [3,14], amalgam [6], and 2-amalgam [12]. All these operations were proved to preserve perfectness but many of them also have interesting polyhedral counterparts. In particular, the knowledge of the polyhedral descriptions of smaller systems yields a description for the composed system in the case of graph substitution [11], clique cutset composition [11], join [13], amalgam [7], and further generalizations [28,24,25]. Given a graph G = (V , E ), a partition (X1 , X2 ) of V is a 2-bond if, for i = 1, 2, there exist nonempty sets Ai , Bi ⊆ Xi such that every node in A1 is adjacent to every node in A2 , every node in B1 is adjacent to every node in B2 and there are no other edges between X1 and X2 .

∗

Corresponding author. Tel.: +39 067716419; fax: +39 067716461. E-mail addresses: [email protected] (A. Galluccio), [email protected] (C. Gentile), [email protected] (P. Ventura).

0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.02.022

A. Galluccio et al. / Discrete Applied Mathematics 161 (2013) 1988–2000

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If the sets Ai and Bi are disjoint and |Xi | ≥ 3, i = 1, 2, the 2-bond is actually the 2-join defined in [31]. If Ai and Bi , i = 1, 2, are (not necessarily disjoint) cliques we speak of a 2-clique-bond. In this paper we introduce a graph composition, named 2-clique-bond composition, that builds a graph G admitting a 2clique-bond starting from two smaller graphs G1 and G2 , each one containing an edge ei with a prescribed property (simplicial edge). We study the stable set polytope of G and we prove that a linear description of STAB(G) can be derived from the linear descriptions of four lower dimensional polytopes: STAB(G1 ), STAB(G1 /e1 ), STAB(G2 ), and STAB(G2 /e2 ) (the graph Gi /ei is obtained from Gi by contracting the edge ei , i = 1, 2). In Section 3 we introduce an auxiliary graph, the W3 -graph, whose stable set polytope is described by the union of the inequalities defining STAB(Gi ) and STAB(Gi /ei ) for i = 1, 2. In Section 4 we explain how to choose facet defining inequalities of STAB(Gi ) and STAB(Gj /ej ) for i, j = 1, 2, i ̸= j, and how to compose them in order to get facet defining inequalities of STAB(G). The main result of the paper (Section 5) states that the inequalities obtained in this way are necessary and sufficient to provide a linear description of STAB(G) when G admits a 2-clique-bond. The proof has been carried out by exploiting the polyhedral properties of the W3 -graphs associated with G1 and G2 . This result can be applied in two different ways: either to build new classes of graphs whose stable set polytope has an explicit linear description or to find the linear description of the stable set polytope of classes of graphs that are known to be decomposable via 2-clique-bonds. In Section 6, we provide examples of both applications. 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 |V | u and v is denoted by uv . A graph G together with a node weighting function w ∈ Q+ is denoted by (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 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 copies of multiple edges). A k-hole Ck = (v1 , v2 , . . . , vk ) is a chordless cycle of length k. A k-wheel Wk = (h : v1 , v2 , . . . , vk ) is a graph consisting of a k-hole Ck = (v1 , v2 , . . . , vk ) and a node h (hub of W ) adjacent to every node of Ck . If k = 3, the 3-wheel W3 is a clique of size 4. A node of V is simplicial if its neighborhood induces a clique. We denote by NG (v) the neighborhood of v in G, i.e., the set of nodes of V adjacent to v ; when there is no confusion we simply write N (v). An edge e = v1 v2 of E is simplicial if N (v1 ) \ {v2 } and N (v2 ) \ {v1 } are two nonempty cliques of G. A clique cutset  of G is a complete subgraph  whose removal disconnects G. Given a vector β ∈ R|V | and a subset U ⊆ V , let β(U ) = i∈U βi . A linear inequality j∈V βj xj ≤ β0 is valid for STAB(G)

if it holds for all x ∈ STAB(G). For short, we also denote a linear inequality β T x ≤ β0 as (β, β0 ). We say that a stable set S is tight for (β, β0 ) if β(S ) = β0 and that S violates (β, β0 ) if β(S ) > β0 . The incidence vector of a stable set that is tight for (β, β0 ) is a root of (β, β0 ). A valid inequality (β, β0 ) for STAB(G) defines a facet of STAB(G) if and only if it is satisfied as an equality by |V | affinely independent roots of (β, β0 ). The facet defining inequalities of STAB(G) constitute the unique nonredundant defining linear system of STAB(G). Given an inequality (β, β0 ) that is valid for STAB(G), the supporting graph Gβ is the subgraph of G induced by the nodes with nonzero coefficients. The nonnegativity inequalities xv ≥0, v ∈ V (G), are known to be facet defining for STAB(G) and we refer to them as trivial inequalities. A linear inequality j∈V πj xj ≤ π0 is said to be a rank inequality of STAB(G) if there exists a subset U ⊆ V such that πi = 1 for each i ∈ U , πi = 0 for each i ∈ V \ U and π0 = α(G[U ]) where G[U ] is the subgraph of G induced by U. A clique inequality (5-hole inequality) is a rank inequality where the subgraph G[U ] is a clique (a 5-hole, respectively). Given 5 a 5-wheel W5 = (h : v1 , v2 , v3 , v4 , v5 ), the inequality i=1 xvi + 2xh ≤ 2 is called 5-wheel inequality. For the sake of completeness, we recall the definition of the sequential lifting procedure defined  in [30] that will be mentioned in the following sections. Let S (G) denote the family of the stable sets of G = (V , E ). If j∈V \{v} βj xj ≤ β0 is a facet defining inequality of STAB(G \ {v}), then the inequality

 j∈V \{v}

βj xj + βv xv ≤ β0 with βv = β0 −

max

S ∈S (G\(N (v)∪{v}))

β(S )

(1)

is facet defining for STAB(G). The inequality (1) is called sequential lifting of j∈V \{v} βj xj ≤ β0 and βv is called the lifting coefficient of v . This procedure can be iterated to generate facet defining inequalities, simply called lifted inequalities, in a higher dimensional space. We denote by SMAX (G, w) the set {S ∈ S (G)|w(S ) = α(G, w)}. Observe that if (γ , γ0 ) is facet defining for STAB(G) then SMAX (G, γ ) is the set of the roots of (γ , γ0 ). The proofs in this paper use basic concepts of (integer) linear programming that we summarize in the following.



Observation 1. Let (γ , γ0 ) be a facet defining inequality of STAB(G). Then, for every valid inequality (β, β0 ) that, up to positive multiplications, is not (γ , γ0 ), there exists a stable set S ∈ SMAX (G, γ ) that is not tight for (β, β0 ) (i.e. β(S ) < β0 ).

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|V |

Observation 2. Let G = (V , E ) be a graph. For any node weighting function w ∈ Q+ , there exists a facet defining inequality (β, β0 ) of STAB(G) such that β(S ) = β0 for any stable set S ∈ SMAX (G, w), i.e., SMAX (G, w) ⊆ SMAX (G, β). A basic result of linear programming (Farkas’ Lemma, e.g., see [27]) 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. |V |

Observation 3. Let G = (V , E ) be a graph, w ∈ Q+ a node weighting function, and ai x ≤ bi , i = 1, . . . , p a system of linear inequalities that are facet defining STAB(G) and such that ai (S ) = bi for any i = 1, . . . , p and any S ∈ SMAX (G, w). If there for p p does not exist λ ∈ R+ such that i=1 λT ai = w , then there exists (β, β0 ) that is facet defining for STAB(G) and has the following properties:

 • there is no λ˜ ∈ Rp+ such that pi=1 λ˜ T ai = β ; • β(S ) = β0 for any S ∈ SMAX (G, w). Chvátal [11] proved the following fundamental result concerning the stable set polytope of graphs containing clique cutsets. Theorem 4. Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs. Let G1 ∪ G2 = (V1 ∪ V2 , E1 ∪ E2 ) and G1 ∩ G2 = (V1 ∩ V2 , E1 ∩ E2 ). If G1 ∩ G2 is a complete graph, then the defining linear system of STAB(G1 ∪ G2 ) is given by the union of the defining linear systems of STAB(G1 ) and STAB(G2 ). This implies that a graph supporting a facet defining inequality cannot contain a clique cutset. 3. W3 -graphs In this section we introduce an auxiliary graph that will be used as an intermediate step to prove our main result. Definition 5. Let H be a graph, let A and B be two cliques of V (H ), and let W3 be the 3-wheel (t∅ : tA , tB , tAB ). The W3 -graph H W = (H , A, B, W3 ) is obtained as follows: V (H W ) = V (H ) ∪ V (W3 ) E (H W ) = E (H ) ∪ E (W3 ) ∪ F1 ∪ F2 ,

where F1 = {uv|u ∈ {tA , tAB }, v ∈ A} and F2 = {uv|u ∈ {tB , tAB }, v ∈ B}.

If G is a graph with a simplicial edge ab, the W3 -graph H W = (H , A, B, W3 ), where H = G \ {a, b}, A = N (a) \ {b}, and B = N (b) \ {a}, is said to be the W3 -graph associated with G. We now study the stable set polytope of W3 -graphs. The first consideration is an easy consequence of Theorem 4. Corollary 6. If H W is a W3 -graph and Dx ≤ d defines STAB(H W \ {t∅ }), then {Dx ≤ d, xtA + xtAB + xtB + xt∅ ≤ 1, xt∅ ≥ 0} defines STAB(H W ). Thus, when H W is a W3 -graph, the only graph supporting a nontrivial facet defining inequality for STAB(H W ) that contains t∅ is the clique W3 itself. In [18] we proved the following. Proposition 7. Let G be a graph and let (β, β0 ) be a nontrivial facet defining inequality of STAB(G). If uv is a simplicial edge of Gβ then βu = βv . This result has some interesting consequences on the coefficients of nodes of W3 in facet defining inequalities of the stable set polytope of W3 -graphs. Indeed, we have the following. Proposition 8. Let H W be a W3 -graph and let (β, β0 ) be a facet defining inequality of STAB(H W ) which is neither a clique inequality nor a nonnegativity inequality. If βtA or βtB is nonzero then βtA = βtB = βtAB . Proof. By Corollary 6, βt∅ = 0 and the edge tA tB is simplicial in every graph supporting a facet defining inequality (β, β0 ) of STAB(H W ) which is neither a clique inequality nor a nonnegativity inequality. Thus, by Proposition 7, tA and tB have the same coefficient, namely βtA = βtB . By Observation 1, there exists a stable set S ∈ SMAX (H W , β) such that S ∩ {A ∪ {tA , tAB }} = ∅, since A ∪ {tA , tAB } is a clique. If t∅ ∈ S, then S \ {t∅ } ∪ {tA } is a stable set violating (β, β0 ), a contradiction. This implies that tB ∈ S, otherwise S ∪ {tA } would contradict the fact that S ∈ SMAX (H W , β). Then βtB = βtA ≥ βtAB because S \ {tB } ∪ {tAB } is a stable set of H W . As (β, β0 ) is facet defining for STAB(H W ), it is not a nonnegativity inequality, and STAB(H W ) has full dimension, then for each node u there exists a stable set that is tight for (β, β0 ) and contains u. In particular, there exists S ∈ SMAX (H W , β) such that tAB ∈ S; then βtAB ≥ βtB = βtA . Hence βtA = βtB = βtAB and the proposition follows.  Given a W3 -graph H W , we say that an inequality (β, β0 ) is a k-inequality, k ∈ {0, 1, 2, 3, 4}, if it is facet defining for STAB(H W ) and it has k nonzero coefficients among {βt∅ , βtA , βtB , βtAB }. Due to Proposition 8 and Corollary 6, we can classify the set of inequalities that define the stable set polytope of H W as follows:

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Observation 9. Let H W = (H , A, B, W3 ) be a W3 -graph and let (β, β0 ) be a nontrivial facet defining inequality of STAB(H W ). Then (β, β0 ) is one of the following: (i) (ii) (iii) (iv) (v) (vi)

a 0-inequality, i.e., βtA = βtB = βtAB = βt∅ = 0, a 1-inequality with βtA = βtB = βt∅ = 0 and βtAB ̸= 0,  the 2-inequality u∈A xu + xtA + xtAB ≤ 1,  the 2-inequality u∈B xu + xtB + xtAB ≤ 1, a 3-inequality with βtA = βtB = βtAB ̸= 0 and βt∅ = 0, the 4-inequality xtA + xtB + xtAB + xt∅ ≤ 1.

Note that inequalities (iii), (iv), and (vi) are clique inequalities. The next lemma explains the connections between the stable set polytopes of the graphs G, G/ab, and the W3 -graph H W associated with G. Lemma 10. Let G be a graph with a simplicial edge e = ab and let z be the node created by the contraction of ab. If H W is the W3 -graph associated with G, then

 βv xv +βˆ xtA +βˆ xtB +βˆ xtAB ≤ β0 with βˆ > 0 is facet defining for STAB(H W ) if and only if v∈V (H ) βv xv +βˆ xa +βˆ xb ≤ β is facet defining for STAB(G) and it is not xa + xb ≤ 1, 0  W ˆ ˆ ˆ 2. v∈V (H ) βv xv + β xtAB ≤ β0 with β > 0 is facet defining for STAB(H ) if and only if v∈V (H ) βv xv + β xz ≤ β0 is facet defining for STAB(G/e) and it is different from the clique inequality supported by A ∪ {z } or B ∪ {z }.

1.



v∈V (H )

Proof. We prove the first part of the lemma by separately considering each direction. Hereafter, we identify tA and tB in H W with a and b in G, respectively. Thus G is an induced subgraph of H W .

βv xv +βˆ xa +βˆ xb ≤ β0 is facet defining for STAB(G), then by applying the sequential lifting procedure described  in (1) to the node tAB , we obtain an inequality v∈V (H ) βv xv + βˆ xtA + βˆ xtB +βtAB xtAB ≤ β0 that is facet defining for STAB(H W ). By Proposition 8, βtAB = βˆ , as claimed. If



v∈V (H )

To prove the other direction, we let



βv xv + βˆ xtA + βˆ xtB + βˆ xtAB ≤ β0 with βˆ > 0

(2)

v∈V (H )

be a 3-inequality of STAB(H W ) and we suppose by contradiction that



βv xv + βˆ xa + βˆ xb ≤ β0

(3)

v∈V (H )

is not facet defining for STAB(G). Then there exists a facet defining inequality



γv xv + γa xa + γb xb ≤ γ0

(4)

v∈V (H )

of STAB(G) that contains all roots of (3). Moreover, the inequality



γv xv + γa xtA + γb xtB + γab xtAB + γ∅ xt∅ ≤ γ0 ,

(5)

v∈V (H )

obtained from (4) by lifting the nodes tAB and t∅ according to (1), is facet defining for STAB(H W ). We distinguish four cases depending on the coefficients γa and γb in (4): Case 1. γa and γb are nonzero and (4) is different from xa + xb ≤ 1. By Proposition 8, γa = γb = γab = γˆ > 0 and, by Corollary 6, γ∅ = 0. We now prove that inequality (5) contains all the roots of (2) and therefore they define the same facet of STAB(H W ). Consequently, (2) is a positive scalar multiple of (5) and the same holds for (3) and (4), thus proving that (3) is also facet defining for STAB(G), contradicting the hypothesis. Let S be a tight stable set for (2). If tAB ̸∈ S then it is straightforward to see that it is also tight for (3), and therefore for (4) and (5). If tAB ∈ S then S ′ = S \ {tAB } ∪ {a} is feasible and tight for (3), and therefore for (4). Thus S is also tight for (5), as claimed. Case 2. Inequality (4) is xa + xb ≤ 1. By Observation 1, there exists a tight stable set S for (2) such that S ∩ V (W3 ) = ∅. Clearly S is tight for (3), but it is not tight for xa + xb ≤ 1. A contradiction.

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(a)

(c)

(b)

Fig. 1. (a) G1 and G2 with simplicial edges a1 b1 and a2 b2 . (b) 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). (c) 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (b2 , a2 ).

Case 3. γa = γb = 0. Suppose first that γab > 0; as STAB(H W ) has full dimension, there exists a stable set S containing tAB that is tight for (2). Then S ′ = S \{tAB }∪{a} is feasible and tight for (3), and therefore for (4), i.e., γ (S ′ ) = γ0 . Hence, γ (S ) = γ (S ′ )−γa +γab > γ0 and S violates (5). A contradiction. It follows that γab = 0 and we can use the same arguments of Case 1 to prove that inequality (3) is facet defining. A contradiction. Case 4. Either γa or γb is zero. Without loss of generality, let γa > 0 and γb = 0. Then, by Observation 9, inequality (5) is the clique inequality supported by A ∪ {tA , tAB }, i.e., γa = γab = 1. Let S be a stable set that is tight for (2). If S intersects A ∪ {tA , tAB } then it is also tight for (5). If S does not intersect A ∪ {tA , tAB } then S contains tB (otherwise S ∪ {tA } would violate (2)). Hence, S is feasible and tight for (3), and therefore for (4), i.e., γ (S ) = γ0 . It follows that, for the stable set S ′ = S \ {tB } ∪ {tA }, γ (S ′ ) = γ (S ) − γb + γa > γ0 . A contradiction. This completes the proof of (1). Finally, (2) follows by observing that G/e is an induced subgraph of H W and that the two inequalities have isomorphic supporting graphs.  If G is a graph with a simplicial edge ab, we call even a facet defining inequality of STAB(G) with nonzero coefficients on a and b that is different from xa + xb ≤ 1; moreover, we call odd a facet defining inequality of STAB(G/ab) with nonzero coefficients on z that is different from the clique inequalities supported by A ∪ {z } or B ∪ {z }, and that is not xz ≥ 0. Thus, according to Lemma 10, there is a one-to-one correspondence between (i) even inequalities of STAB(G) and 3-inequalities of STAB(H W ), (ii) odd inequalities of STAB(G/ab) and 1-inequalities of STAB(H W ). 4. 2-clique-bond composition We define a graph composition that builds graphs admitting a 2-clique-bond. Definition 11. Let G1 and G2 be two disjoint graphs. Let (ai , bi ) be an ordered pair of nodes such that ai bi is a simplicial edge of Gi and let Ai = N (ai ) \ {bi } and Bi = N (bi ) \ {ai }, i = 1, 2. The 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ) is the graph G obtained by deleting the nodes ai and bi , for i = 1, 2, and joining every node in A1 with every node in A2 and every node of B1 with every node of B2 . In Fig. 1, two examples of 2-clique-bond composition are shown. Note that different orders of pairs lead to different graphs after applying the 2-clique-bond composition. In the following we see how to obtain valid inequalities for the stable set polytope of a graph G that is the 2-clique-bond composition of G1 and G2 . Definition 12. Let G be the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Let zi be the node resulting from the contraction of ai bi , i = 1, 2, in Gi /ai bi . j Let β i x ≤ β0i be an even inequality of STAB(Gi ) and let β j x ≤ β0 be an odd inequality of STAB(Gj /aj bj ) such that

βai i = βbi i = βzjj = 1, for i, j ∈ {1, 2} and i ̸= j.

j

An inequality is said to be an even–odd combination of (β i , β0i ) and (β j , β0 ) if it has the following form:

 v∈V (Gi \{ai ,bi })

βvi xv +

 v∈V ((Gj /aj bj )\{zj })

βvj xv ≤ β0i + β0j − 1.

(6)

j

Note that the conditions βai i = βbi i = βzj = 1 are not restrictive. In fact, by Proposition 7, βai i = βbi i . Therefore it is always j

j

possible to normalize (β i , β0i ) and (β j , β0 ) so that βai i = βbi i = βzj = 1.

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Fig. 2. Above, G is the 2-clique-bond composition of H and W5 along (a1 , b1 ) and (a2 , b2 ). Below, a facet defining inequality of STAB(G) obtained as the even–odd combination of an odd inequality of STAB(H /a1 b1 ) and an even inequality of STAB(W5 ).

Lemma 13. Let G be the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Let (β i , β0i ) be an odd inequality j

of STAB(Gi /ai bi ) and (β j , β0 ) an even inequality of STAB(Gj ), with i, j ∈ {1, 2}, i ̸= j. If (β, β0 ) is the even–odd combination j 0

of (β , β ) and (β , β ) then it is valid for STAB(G). i

i 0

j

Proof. Without loss of generality, let i = 1 and j = 2. To reach a contradiction, let S be a maximal stable set of G that violates (β, β0 ), i.e., β(S ) = β 1 (S ∩ V (H1 )) + β 2 (S ∩ V (H2 )) > β01 + β02 − 1, where Hi = Gi \ {ai , bi } for i = 1, 2. Clearly |S ∩ (A1 ∪ A2 ∪ B1 ∪ B2 )| ≤ 2. Suppose first that S ∩ A2 = ∅. As (β 1 , β01 ) is valid for STAB(G1 /a1 b1 ), and S ∩ V (H1 ) is a stable set of G1 , β 1 (S ∩ V (H1 )) ≤ β01 and, consequently, β 2 (S ∩ V (H2 )) > β02 − 1. But then, as βa22 = 1, (S ∩ V (H2 ))∪{a2 } is a stable set of G2 that violates (β 2 , β02 ), a contradiction. So S ∩ A2 ̸= ∅ and symmetric arguments also prove that S ∩ B2 ̸= ∅. As a consequence, S ∩ A1 = S ∩ B1 = ∅. Since (β 2 , β02 ) is valid for STAB(G2 ) and S ∩ V (H2 ) is a stable set of G2 , we have that β 2 (S ∩ V (H2 )) ≤ β02 and then β 1 (S ∩ V (H1 )) > β01 − 1. But then (S ∩ V (H1 )) ∪ {z1 } would be a stable set of G1 /a1 b1 violating (β 1 , β01 ), a contradiction.  In Fig. 2 it is shown how to obtain a valid inequality for the stable set of the graph G as an even–odd combination of an even inequality of STAB(W5 ) (the 5-wheel inequality) and an odd inequality of STAB(H /a1 b1 ) (where a1 b1 is the simplicial edge involved in the 2-clique-bond composition depicted). Note that the coefficients of the odd inequality have been multiplied by 1/2 so that z1 has coefficient 1. 5. The main result In this section we explain how to find a linear description of STAB(G) when G admits a 2-clique-bond once we know the linear description of the stable set polytopes of four smaller graphs associated with G. To this aim we need some properties of the maximal weighted stable sets of the W3 -graphs associated with the graphs involved in the 2-clique-bond composition. Definition 14. Let G be obtained as the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Let HiW = (Hi , Ai , Bi , W3i ) be the W3 -graphs associated with Gi . For any nonnegative node weighting function w of G, we define the generating weighting functions w i on V (HiW ) as follows

 w(u),    α(Hj \ Bj , w) w i (u) = α(Hj \ Aj , w),    α(Hj \ (Aj ∪ Bj ), w), α(Hj , w),

if u if u if u if u if u

∈ V (Hi ) = tAi = tBi = t∅i i = tAB

with i, j = 1, 2 and i ̸= j. We also say that the weighted graph (G, w) is generated by the weighted graphs (H1W , w 1 ) and (H2W , w2 ).

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i In other words, we assign weights to the nodes t∅i , tAi , tBi , and tAB of the W3 -graph associated with Gi to encode all the i information about the maximum weight stable sets of G. Observe that w i (t∅i ) ≤ w i (ui ) ≤ w i (tAB ), with ui ∈ {tAi , tBi }, for W W 1 2 i = 1, 2 and that α(G, w) = α(H1 , w ) = α(H2 , w ). i To simplify the notation, in the remainder of this section, we write wAi , wBi , wAB , and w∅i to represent the value of a vector i i i i i w on the nodes tA , tB , tAB , and t∅ , respectively.

Definition 15. Let w be a node weighting function for G and let w i be the corresponding generating weighting functions on V (HiW ), i = 1, 2. Two stable sets S1 ∈ SMAX (H1W , w 1 ) and S2 ∈ SMAX (H2W , w 2 ) are said to generate S ∈ SMAX (G, w) if S = S1 ∪ S2 \ (V (W31 ) ∪ V (W32 )). Conversely, given a stable set S ∈ SMAX (G, w), it is possible to produce S1 ∈ SMAX (H1W , w 1 ) and S2 ∈ SMAX (H2W , w 2 ) generating S as follows: for i, j = 1, 2 and i ̸= j, let Si ∩ V (Hi ) = S ∩ V (Hi ) and

i t ,    Ai t , i Si ∩ {tAi , tBi , tAB , t∅i } = Bi t ,    AB t∅i ,

if S if S if S if S

∩ Aj ∩ Aj ∩ Aj ∩ Aj

̸ ∅, = = ∅, ̸= ∅, = ∅,

S S S S

∩ Bj ∩ Bj ∩ Bj ∩ Bj

=∅ ̸= ∅ ̸= ∅ = ∅.

If S1 ∈ SMAX (H1W , w 1 ) and S2 ∈ SMAX (H2W , w 2 ) generate S ∈ SMAX (G, w), then the following conditions hold:

w(S ) = α(G, w) = w(S1 ∩ V (H1 )) + w(S2 ∩ V (H2 )) = w 1 (S1 ) = α(H1W , w1 ) = w(S1 ∩ V (H1 )) + w1 (S1 ∩ V (W31 )) = w 2 (S2 ) = α(H2W , w2 ) = w(S2 ∩ V (H2 )) + w2 (S2 ∩ V (W32 )) = w1 (S1 ∩ V (W31 )) + w 2 (S2 ∩ V (W32 )). Lemma 16. Let G be the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Let (γ 1 , γ01 ) be a 1-inequality of STAB(H1W ) and (γ 2 , γ02 ) be a 3-inequality of STAB(H2W ), where H1W and H2W are the W3 -graphs associated with G1 and G2 , respectively. Then γ 1 (S1 ∩ V (W31 )) + γ 2 (S2 ∩ V (W32 )) ≥ 1 for each pair of stable sets S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) generating a maximal stable set S of G. Proof. As S1 and S2 are tight for (γ 1 , γ01 ) and (γ 2 , γ02 ), respectively, then

γ 1 (S1 ) = γ 1 (S1 ∩ V (H1 )) + γ 1 (S1 ∩ V (W31 )) = γ01 ,

(7)

γ 2 (S2 ) = γ 2 (S2 ∩ V (H2 )) + γ 2 (S2 ∩ V (W32 )) = γ02 .

According to Lemma 10, let (β 1 , β01 ) be the odd inequality of STAB(G1 /a1 b1 ) corresponding to (γ 1 , γ01 ) and let (β 2 , β02 ) be the even inequality of STAB(G2 ) corresponding to (γ 2 , γ02 ). As γ 1 (V (H1 )) = β 1 (V (H1 )), γ01 = β01 , γ 2 (V (H2 )) = β 2 (V (H2 )), and γ02 = β02 , by Lemma 13 we have that

γ 1 (S1 ∩ V (H1 )) + γ 2 (S2 ∩ V (H2 )) ≤ γ01 + γ02 − 1.

(8)

The lemma follows by adding equalities (7) and subtracting inequality (8).



It is easy to prove the following. Observation 17. If Si ∈ SMAX (HiW , w i ), then there always exists a stable set Sj ∈ SMAX (HjW , w j ) such that Si and Sj generate a stable set S ∈ SMAX (G, w), for i, j ∈ {1, 2} and j ̸= i. Moreover, the following property holds. Lemma 18. Let (G, w) be generated by (H1W , w 1 ) and (H2W , w 2 ). Furthermore, let Si ∈ SMAX (HiW , w i ) and Sj ∈ SMAX (HjW , w j ) j

j

j

j

j

i generate S ∈ SMAX (G, w). If tAB ∈ Si and u ∈ Sj , with u ∈ {tA , tB , tAB }, i, j ∈ {1, 2} and j ̸= i, then wu = w∅ . j

j

j

i Proof. We first consider u = tA . Let S ′ = Si \ {tAB } and then assume, by contradiction, that wA > w∅ . It follows that there j

exists a stable set S in Hi \ Bi such that w (S ) > w∅ = α(Hi \ (Ai ∪ Bi ), w ) = w (S ). But then (S \ S ′ ) ∪ S ′′ would be a stable set of G whose weight is greater than w(S ), contradicting the fact that S ∈ SMAX (G, w). Similar arguments prove the j j cases u = tB and u = tAB .  ′′

i

′′

i

i

′

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 four graphs: G1 , G1 /e1 , G2 , and G2 /e2 .

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Theorem 19. Let Gi be a graph with a simplicial edge ei = ai bi , i = 1, 2 and let G be the graph obtained as the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Then STAB(G) is described by the following inequalities:

• • • •

nonnegativity inequalities; facet defining inequalities of STAB(Gi ) with zero coefficients on the endnodes of ei for i = 1, 2; clique inequalities induced by A1 ∪ A2 and B1 ∪ B2 ; even–odd combinations of facet defining inequalities of STAB(Gi ) and STAB(Gj /aj bj ) for each i, j = 1, 2 and i ̸= j.

Proof. Suppose by contradiction that there exist a graph G and an inequality (γ , γ0 ) 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 , γ0 ) is fully supported by G. Let H1W = (H1 , A1 , B1 , W31 ) and H2W = (H2 , A2 , B2 , W32 ) be the W3 -graphs associated with G1 and G2 , according to Definition 5. Now consider the vector γ as a node weighting function of G and let γ i be the generating weighting functions γ i of γ on V (HiW ), i = 1, 2, as in Definition 14. By Observation 2, there exists a facet defining inequality (β i , β0i ) of STAB(HiW ) such that all stable sets in SMAX (HiW , γ i ) are tight for (β i , β0i ), i = 1, 2. Moreover, since HiW is a W3 -graph, the inequality (β i , β0i ), i = 1, 2, is either a nonnegativity inequality or an inequality of type (i), . . . , (vi) of Observation 9. As (γ , γ0 ) is fully supported by G, there is at least one node ui ∈ V (Hi ) with γuii > 0 for i = 1, 2. This implies that the

vector γ i , i = 1, 2, cannot be obtained as a conic combination of a set of facet defining inequalities of STAB(HiW ) containing only nonnegativity inequalities and inequalities of type vi). By Observation 3, this implies that there exists a facet defining inequality (β i , β0i ), i = 1, 2, that is not a nonnegativity inequality nor an inequality of type (vi) and such that all stable sets in SMAX (HiW , γ i ) are tight for (β i , β0i ), i = 1, 2. Hence, we may assume that (β 1 , β01 ) and (β 2 , β02 ) are restricted to be of type (i), . . . , (v) and we examine all possibilities for (β 1 , β01 ) and (β 2 , β02 ) in the following six cases. In each case we consider a tight stable set S for (γ , γ0 ) together with two stable sets S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) that generate S and that are tight for (β 1 , β01 ) and (β 2 , β02 ), respectively. We then properly modify S1 and S2 to obtain contradictions. Case 1. (β i , β0i ) is a 0-inequality for some i ∈ {1, 2}. Without loss of generality let i = 1, i.e., βu1 = 0 for each u ∈ V (W31 ). Let (β, β0 ) be an inequality of STAB(G) whose coefficients βv are equal to βv1 if v ∈ V (H1 ) and zero otherwise, and β0 = β01 , i.e. we consider the same inequality in G. Then, let S ∈ SMAX (G, γ ), and let S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) generate S: as S ∩ V (H1 ) = S1 ∩ V (H1 ) and S1 is tight for (β 1 , β01 ) by assumption, S satisfies (β, β0 ) as an equality. As this holds for any S ∈ SMAX (G, γ ), (γ , γ0 ) is a positive multiple of (β, β0 ), so contradicting the hypothesis that (γ , γ0 ) is not a positive multiple of an inequality listed in the thesis. Case 2. (β i , β0i ) is a 2-inequality of type (iii) or (iv) for some i ∈ {1, 2}. Without loss of generality, let i = 1 and let (β 1 , β01 ) be of type (iii). Let S ∈ SMAX (G, γ ) with S ∩ (A1 ∪ A2 ) = ∅ (such a stable set exists because of Observation 1 applied to the clique inequality induced by the nodes of A1 ∪ A2 ) and let S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) generate S. As S1 ∈ SMAX (H1W , γ 1 ) and (β 1 , β01 ) is of type (iii), then 1 1 S1 ∩ (A1 ∪ {tA1 , tAB }) ̸= ∅, i.e. there exists u ∈ {tA1 , tAB } contained in S1 . 1 1 If u = tAB , then S1 ∩ (A1 ∪ B1 ) = ∅. As S ∩ A2 = ∅, γ (S2 ∩ V (H2 )) = γB1 . Moreover, the stable set S ∗ = S1 \ {tAB } ∪ {tB1 } 1 W 1 1 1 1 is not tight for (β , β0 ) and, as by hypothesis every stable set in SMAX (H1 , γ ) is tight for (β , β0 ), it follows that S ∗ 1 > γB1 , and so there exists a stable set S ′ of H2 such that does not belong to SMAX (H1W , γ 1 ). As a consequence, γAB

γ (S ′ ) > α(H2 \ A2 , γ ) = γB1 = γ (S2 ∩ V (H2 )). But then S˜ = (S ∩ V (H1 )) ∪ S ′ is a stable set (because S ∩ (A1 ∪ B1 ) = ∅) with γ (S˜ ) > γ (S ), thus contradicting the fact that S ∈ SMAX (G, γ ). 1 If u = tA1 , then we may assume that S1 ∩ B1 ̸= ∅ since otherwise S1 \ {tA1 } ∪ {tAB } ∈ SMAX (H1W , γ 1 ) and we are in the 1 previous case. Hence, γ (S2 ∩ V (H2 )) = γ∅ and we can use arguments similar to the above ones to prove that γA1 > γ∅1 . This implies that there exists a stable set S ′′ of H2 \ B2 such that γ (S ′′ ) > α(H2 \ (A2 ∪ B2 ), γ ) = γ∅1 . As S¯ = (S1 ∩ V (H1 )) ∪ S ′′ is a stable set (because S1 ∩ A1 = ∅) with γ (S¯ ) > γ (S ), it contradicts again the fact that S ∈ SMAX (G, γ ). Case 3. (β 1 , β01 ) and (β 2 , β02 ) are both 1-inequalities. Let S ∈ SMAX (G, γ ) be such that S ∩(A1 ∪A2 ) = ∅ (such a stable set exists by Observation 1). Observe that |S ∩(B1 ∪B2 )| ≤ 1 and assume, without loss of generality, that S ∩ B1 = ∅. Let S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) be two stable sets 1 1 generating S. In particular, S1 is tight for the 1-inequality (β 1 , β01 ) and it is not difficult to see that tAB ∈ S1 . Indeed, if tAB ̸∈ S1 , 1 1 1 1 1 1 then S1 \ {tA , tB , t∅ } ∪ {tAB } violates (β , β0 ), a contradiction. 1 1 If γu1 = γAB for some u ∈ {tA1 , tB1 , t∅1 }, then S ′ = S1 \ {tAB } ∪ {u} would be a maximum stable set in SMAX (H1W , γ 1 ) such 1 1 1 1 ′ that β (S ) = β0 − βAB < β0 , violating the hypothesis that all the stable sets in SMAX (H1W , γ 1 ) are tight for (β 1 , β01 ). 1 Hence γu1 < γAB for u ∈ {tA1 , tB1 , t∅1 }. This implies that there exists a stable set S ′′ ⊆ V (H2 ) such that γ (S ′′ ) = α(H2 , γ ) = 1 1 γAB > γB = α(H2 \ A2 , γ ) = γ (S ∩ V (H2 )). But then, as S ∩ A1 = ∅ and S ∩ B1 = ∅, S˜ = (S ∩ V (H1 )) ∪ S ′′ would be a stable set of G with γ (S˜ ) > γ (S ), violating the fact that S ∈ SMAX (G, γ ). A contradiction.

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Case 4. (β 1 , β01 ) and (β 2 , β02 ) are both 3-inequalities. As for Case 3, let S ∈ SMAX (G, γ ) be such that S ∩ (A1 ∪ A2 ) = ∅. Since |S ∩ (B1 ∪ B2 )| ≤ 1 we can assume without loss 1 of generality that S ∩ B1 = ∅. Let S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) be two stable sets generating S. As γAB ≥ γu1

1 for any u ∈ {tA1 , tB1 , t∅1 }, there exists a stable set S˜1 ∈ SMAX (H1W , γ 1 ) with tAB ∈ S˜1 and S˜1 ∩ V (H1 ) = S1 ∩ V (H1 ). Moreover,

2 2 as γAB ≥ γu2 ≥ γ∅2 for any u ∈ {tA2 , tB2 }, there exists a stable set S˜2 ∈ SMAX (H2W , γ 2 ) containing v ∈ {tA2 , tAB } and such that

S˜2 ∩ V (H2 ) = S2 ∩ V (H2 ). Now observe that S˜1 and S˜2 do generate S. Furthermore, as S˜2 is tight for (β 2 , β02 ) and S˜2 \ {v} ∪ {t∅2 } is not tight for (β 2 , β02 ), γv2 > γ∅2 , contradicting Lemma 18.

Case 5. (β 1 , β01 ) is a 1-inequality and (β 2 , β02 ) is a 3-inequality (or vice-versa). 1 2 By Proposition 8, we can normalize (β 1 , β01 ) and (β 2 , β02 ) so that βAB = βA2 = βB2 = βAB = 1. Let (β, β0 ) be the inequality

 v∈V (H1 )

βv1 xv +

 v∈V (H2 )

βv2 xv ≤ β01 + β02 − 1.

(9)

Thus (β, β0 ) is the even–odd combination of an odd inequality of STAB(G1 /e1 ) and an even inequality of STAB(G2 ). We now show that (β, β0 ) is satisfied as an equality by each S ∈ SMAX (G, γ ), thus contradicting the hypothesis that (γ , γ0 ) is not a positive multiple of an inequality listed in the thesis. Suppose conversely that there exists a stable set S ∈ SMAX (G, γ ) 1 2 such that β(S ∩ V (H1 )) + β(S ∩ V (H2 )) < β0 = β01 + β02 − 1. As βAB = 1 and βAB = βA2 = βB2 = 1, this implies that 1 2 2 2 S is not tight for (β, β0 ) if and only if tAB ∈ S1 and u ∈ S2 , where u ∈ {tA , tB , tAB } and, as usual, S1 ∈ SMAX (H1W , γ 1 ) and S2 ∈ SMAX (H2W , γ 2 ) generate S. As by hypothesis every stable set in SMAX (H2W , γ 2 ) is tight for (β 2 , β02 ), it follows that γu2 > γ∅2 , so contradicting Lemma 18. All cases lead to contradictions and the theorem follows.  The next theorem shows that the description of STAB(G) given by Theorem 19 is also minimal. Theorem 20. Let G be the 2-clique-bond composition of G1 and G2 along (a1 , b1 ) and (a2 , b2 ). Let (β i , β0i ) be an odd inequality j

of STAB(Gi /ai bi ) and (β j , β0 ) an even inequality of STAB(Gj ), with i, j ∈ {1, 2}, i ̸= j. If (β, β0 ) is the even–odd combination j 0

of (β i , β0i ) and (β j , β ) then it is facet defining for STAB(G).

Proof. Without loss of generality, let i = 1 and j = 2. Suppose by contradiction that (β, β0 ) is not facet defining for STAB(G); then there exists an inequality (π , π0 ) that defines a facet of STAB(G) containing all the roots of (β, β0 ). By Theorem 19, (π, π0 ) is one of the following:

• • • •

nonnegativity inequalities; facet defining inequalities of STAB(Gi ) with zero coefficients on the endnodes of ei for each i = 1, 2; clique inequalities induced by A1 ∪ A2 or B1 ∪ B2 ; even–odd combination of either an even inequality of STAB(G1 ) and an odd inequality of STAB(G2 /e2 ), or an odd inequality of STAB(G1 /e1 ) and an even inequality of STAB(G2 ). Let H1W = (H1 , A1 , B1 , W31 ) and H2W = (H2 , A2 , B2 , W32 ) be the W3 -graphs associated with G1 and G2 . According to

Lemma 10, let (βˆ 1 , β01 ) and (βˆ 2 , β02 ) be the facet defining inequalities for STAB(H1W ) and STAB(H2W ) that correspond to (β 1 , β01 ) of STAB(G1 /e1 ) and to (β 2 , β02 ) of STAB(G2 ), respectively. First consider the case where (π , π0 ) is a nonnegativity inequality or a facet defining inequality of STAB(Gi ) with zero coefficients on the endnodes of ei for some i ∈ {1, 2} and, without loss of generality, assume that it is defined over the nodes of H1 . Then the inequality (π 1 , π01 ) of STAB(H1W ) that has the same coefficients as (π , π0 ) on V (H1 ), zero otherwise, and

π01 = π0 is facet defining for STAB(H1W ). Moreover, by Observation 1, there exists S1 ∈ SMAX (H1W , βˆ 1 ) such that S1 is not tight for (π 1 , π01 ). By Observation 17 there exists also S2 ∈ SMAX (H2W , βˆ 2 ) such that S1 and S2 generate S ∈ SMAX (G, β). By construction, such S is not tight for (π , π0 ), contradicting the assumption. Suppose now that (π , π0 ) is the clique inequality induced by A1 ∪ A2 . Then again, as (β 1 , β01 ) is facet defining for 1 }; STAB(H1W ), there exists S1 ∈ SMAX (H1W , βˆ 1 ) such that S1 is not tight for the clique inequality defined by A1 ∪ {tA1 , tAB 1 1 W ˆ2 as a consequence S1 ∩ {tB , t∅ } ̸= ∅. By Observation 17, there exists S2 ∈ SMAX (H2 , β ) such that S1 and S2 generate S ∈ SMAX (G, β). As tB1 ∈ S1 or t∅1 ∈ S1 we can choose S2 such that S2 ∩ A2 = ∅. As a consequence, S is not tight for (π , π0 ), a contradiction. Symmetric arguments prove the case where (π , π0 ) is the clique inequality induced by B1 ∪ B2 . Consider now the case when (π , π0 ) is an even–odd combination of an odd inequality of STAB(G1 /e1 ) and an even inequality of STAB(G2 ). Let (π 1 , π01 ) be the corresponding 1-inequality that is facet defining for STAB(H1W ) and (π 2 , π02 ) be the corresponding 3-inequality that is facet defining for STAB(H2W ), respectively. If (π 1 , π01 ) is equivalent to (βˆ 1 , β01 ) and (π 2 , π02 ) is equivalent to (βˆ 2 , β02 ), then (π , π0 ) is equivalent to (β, β0 ) and we get a contradiction. Then assume that (π 1 , π01 ) cannot be obtained as a positive multiple of (βˆ 1 , β01 ). Hence, there is S1 ∈ SMAX (H1W , βˆ 1 ) such that π 1 (S1 ) < π01 . Again, by Observation 17 there is S2 ∈ SMAX (H2W , βˆ 2 ) such that S1 and S2 generate S ∈ SMAX (G, β). As π 2 (S2 ) ≤ π02 , then

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it is not difficult to see that, in order to have S tight for (π , π0 ), π 1 (S1 ∩ V (W31 )) + π 2 (S2 ∩ V (W32 )) < 1, contradicting Lemma 16. The case when (π 2 , π02 ) is not equivalent to (βˆ 2 , β02 ) can be similarly proved. Analogous arguments apply to the case where (π , π0 ) is an even–odd combination of an even inequality of STAB(G1 ) and odd inequality of STAB(G2 /e2 ).  By Theorem 20 it is not difficult to verify that the graph G in Fig. 2 supports a facet defining inequality obtained as the even–odd combination of the 5-wheel inequality of STAB(W5 ) and the facet defining inequality of STAB(H /e) defined by Giles and Trotter in [21]. 6. Iterated 2-clique-bond compositions Up to this point we have considered only graphs that are obtained by performing a single 2-clique-bond composition of a given pair of graphs G1 and G2 . In this section we extend our results to graphs obtained by repeated applications of the 2-clique-bond composition. As the stable set polytope of graphs containing clique cutsets has been extensively studied by Chvátal [11], we restrict our attention to simplicial edges that do not lead to clique cutsets. More formally, we have the following. Definition 21. Given a graph H, a simplicial edge ab ∈ E (H ) is strongly simplicial if the following holds:

(∗) (N (a) \ {b}) ̸⊆ (N (b) \ {a}) and (N (b) \ {a}) ̸⊆ (N (a) \ {b}). Note that if (∗) is not satisfied by a simplicial edge, say a1 b1 , the graph G obtained as the 2-clique-bond of G1 and G2 along (a1 , b1 ) and any other pair (a2 , b2 ) associated with a simplicial edge a2 b2 of G2 contains the clique cutset (N (a1 ) ∪ N (b1 )) \ {a1 , b1 }. Definition 22. Let H be a graph and let ΓH = {ei = ai bi , i = 1, . . . , k} be a set of pairwise nonincident strongly simplicial edges of H. Let T be a family of graphs such that each graph Ti in T contains a strongly simplicial edge fi = ui vi , i = 1, . . . , k. A graph Hk obtained by k applications of the 2-clique-bond composition to H is defined as follows:

• H0 = H, • Hi is the 2-clique-bond composition of Hi−1 and Ti along (ai , bi ) and (ui , vi ), i = 1, . . . , k. Given two families of graphs H and T , the graphs Hi , i = 1, . . . , k, obtained from a graph H ∈ H by repeated applications of 2-clique-bond composition with graphs in T is denoted by G(H , T ). Note that in Definition 22 the 2-clique-bond compositions are performed along pairs of nodes corresponding to strongly simplicial edges of Hi−1 that are also strongly simplicial in the original graph H. In fact, we have the following proposition: Proposition 23. Let Hi ∈ G(H , T ) be obtained after i 2-clique-bond compositions according to Definition 22 and let ΓHi = {aj bj : j > i}. Then ΓHi is a matching defined by strongly simplicial edges in Hi for each i ∈ {1, . . . , k}. Proof. Clearly, the edges in ΓHi form a matching in Hi . To prove that aj bj is strongly simplicial in Hi for j > i, we suppose, by induction, that aj bj is strongly simplicial in Hi−1 . If aj and bj are adjacent to neither ai nor bi , then NHi (aj ) = NHi−1 (aj ) and NHi (bj ) = NHi−1 (bj ) and the thesis follows. Claim 1. Neither aj nor bj belongs to NHi (ai ) ∩ NHi (bi ). Suppose conversely that aj is adjacent to ai and bi . As ai bi is strongly simplicial in Hi−1 , there exists x adjacent to ai and not to bi . Since (NHi−1 (ai ) \ {bi }) is a clique, x is adjacent to aj . If x ̸= bj , then {x, bi } is a stable set in NHi−1 (aj ) \ {bj }, contradicting that aj bj is simplicial in Hi−1 . Then x = bj . Since ai bi is strongly simplicial, there exists y ̸= x adjacent to bi and not to ai ; thus, {y, ai } is a stable set in NHi−1 (aj ) \ {bj }, contradicting that aj bj is simplicial in Hi−1 . Hence, the claim follows for aj . By exchanging aj with bj in the above proof, we can prove that bj is not adjacent to both ai and bi .  Without loss of generality, assume that aj is adjacent to ai , j > i. By Claim 1, aj is not adjacent to bi . By exchanging the role of i and j, we can prove that bj is not adjacent to ai . We distinguish two cases. Case 1. bj is not adjacent to bi . After the 2-clique-bond composition of Hi−1 and Ti along (ai , bi ) and (ui , vi ), the adjacencies of aj and bj are as follows:

• NHi (aj ) = (NHi−1 (ai ) \ {aj , bi }) ∪ (NTi (ui ) \ {vi }) ∪ {bj }, • NHi (bj ) = NHi−1 (bj ). By Definition 11, NHi (aj ) \ {bj } is a clique while NHi (bj ) \ {aj } is a clique by induction hypothesis. Thus, aj bj is simplicial. To prove that it is also strongly simplicial, consider a node u∗ ∈ (NTi (ui ) \ {vi }) \ (NTi (vi ) \ {ui }) (it exists because ui vi is a

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strongly simplicial edge of Ti ). By Definition 11, u∗ is adjacent to aj and not to bj in Hi . As aj bj is strongly simplicial in Hi−1 and bi ̸∈ NHi−1 (bj ), there exists in Hi a node y adjacent to bj and not to aj . Hence, condition (∗) of Definition 21 is satisfied by aj bj in Hi , as claimed. Case 2. bj is adjacent to bi . After the 2-clique-bond composition of Hi−1 and Ti along (ai , bi ) and (ui , vi ), the adjacencies of aj and bj are as follows:

• NHi (aj ) = (NHi−1 (ai ) \ {aj , bi }) ∪ (NTi (ui ) \ {vi }) ∪ {bj }, • NHi (bj ) = (NHi−1 (bi ) \ {bj , ai }) ∪ (NTi (vi ) \ {ui }) ∪ {aj }. As in the previous case, NHi (aj ) \ {bj } and NHi (bj ) \ {aj } are both cliques, thus implying that aj bj is simplicial. Furthermore, let u∗ ∈ (NTi (ui ) \ {vi }) \ (NTi (vi ) \ {ui }) and v ∗ ∈ (NTi (vi ) \ {ui }) \ (NTi (ui ) \ {vi }) (they exist since ui vi is strongly simplicial). So, in Hi , u∗ is adjacent to aj and not to bj , and v ∗ is adjacent to bj and not to aj . Thus, aj bj is strongly simplicial.  As a polyhedral counterpart of Definition 22, we now define a family of inequalities obtained by repeatedly performing even–odd combinations of inequalities. Definition 24. Let L be a family of facet defining inequalities for the stable set polytope containing the clique inequalities. An inequality (γ , γ0 ) ∈ L∗ if and only if either (γ , γ0 ) ∈ L, or (γ , γ0 ) satisfies the following: • it is supported by a graph G obtained as the 2-clique-bond composition of G1 and G2 along the pairs (a1 , b1 ) and (a2 , b2 ), and • it is the even–odd combination of two inequalities in L∗ that are facet defining for STAB(Gi ) and STAB(Gj /aj bj ), with i, j ∈ {1, 2} and i ̸= j. We say that L is closed under even–odd combination if L = L∗ . As an example, consider as L the family of rank inequalities that are facet defining for the stable set polytope and denote it by R. Lemma 25. The family R is closed under even–odd combination. Proof. To prove the lemma, it suffices to show that the even–odd combinations of rank inequalities that are facet defining produce rank inequalities. Let G be a graph obtained as the 2-clique-bond composition of two graphs G1 and G2 along the pairs (a1 , b1 ) and (a2 , b2 ). Let (β, β0 ) be obtained as the even–odd combination of two rank inequalities: (β i , β0i ), facet defining for STAB(Gi ), and

(β j , β0j ), facet defining for STAB(Gj /aj bj ), for i, j ∈ {1, 2} and j ̸= i. Moreover, let Uh ⊆ V (Gh ) such that u ∈ Uh if and only if βuh = 1, for h = 1, 2. By Definition 12, βv = 1 for each v ∈ (Ui ∪ Uj \ {ai , bi , zj }) and βv = 0 for each v ∈ (V (G) \ (Ui ∪ Uj ∪ {ai , bi , zj })). Moreover, by Theorem 20, β0 = α(G[Ui ∪ Uj \ {ai , bi , zj }]). 

Given a set L of facet defining inequalities of STAB(G), we consider the polyhedron L∗ STAB(G) = {x ∈ RV+ |x satisfies L∗ } and call a graph L∗ -perfect if L∗ STAB(G) = STAB(G) (see also [22]). An immediate consequence of Theorem 19 is the following. Corollary 26. Let G be the 2-clique-bond of G1 and G2 along the pairs (a1 , b1 ) and (a2 , b2 ). If Gi and Gi /ai bi , 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. Proof. Theorem 19 states that any nontrivial facet defining inequality (β, β0 ) needed to describe STAB(G) is either a facet defining inequality of STAB(Gi ) with zero coefficient on ai , bi , i = 1, 2, or a clique inequality, or an even–odd combination of facet defining inequalities of STAB(Gi ) and STAB(Gj /aj bj ) for i, j ∈ {1, 2}, i ̸= j. As, by hypothesis, Gi and Gi /ai bi , i = 1, 2 are L-perfect, every nontrivial facet defining inequality of STAB(Gi ) and STAB(Gi /ai bi ) is in L. By Definition 24, this implies that (β, β0 ) is in L∗ .  Given a set L, the above result can be applied iteratively to obtain classes of graphs that are L∗ -perfect. In the remainder we denote by H /F the graph obtained from H by contracting all the edges of F ⊆ ΓH . Theorem 27. Let H and T be two families of graphs. Let ΓH denote a set of pairwise nonincident strongly simplicial edges of a graph H ∈ H . If H and H /F are L-perfect for each H ∈ H and for each F ⊆ ΓH , and T and T /uv are L-perfect for each T ∈ T and uv strongly simplicial edge, then every graph in G(H , T ) is L∗ -perfect. Moreover, if L is closed under even–odd combinations, then every graph in G(H , T ) is L-perfect.

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1999

Proof. According to Definition 22, every graph in G(H , T ) is obtained by iteratively applying the 2-clique-bond composition to a graph H ∈ H with graphs in T . So, let Hk ∈ G(H , T ) be obtained from H by k applications of the 2-clique-bond composition with graphs Ti ∈ T , i = 1, . . . , k, along pairs (ai , bi ) and (ui , vi ) such that ΓH = {ei = ai bi , i = 1, . . . , k} is a set of pairwise nonincident strongly simplicial edges of H and fi = ui vi is a strongly simplicial edge of Ti for i = 1, . . . , k. The proof is by induction on k. If k = 1, then H1 is L∗ -perfect by Corollary 26. Let k > 1 and let Hk be the 2-clique-bond composition of Hk−1 and Tk along (ak , bk ) and (uk , vk ). By observing that (L∗ )∗ = L∗ and by using Corollary 26, in order to prove that Hk is L∗ -perfect it suffices to show that the graphs Tk , Tk /fk , Hk−1 , and Hk−1 /ek are L∗ -perfect. Now, Tk and Tk /fk are L-perfect by hypothesis, and Hk−1 is L∗ -perfect by induction hypothesis. Hence, it remains to show that Hk−1 /ek is L∗ -perfect. As the edges {e1 , e2 , . . . , ek } are pairwise nonincident, Hk−1 /ek can be obtained by k − 1 iterated 2-clique-bond compositions of T1 , T2 , . . . , Tk−1 to the graph H /ek along the pairs (a1 , b1 ), . . . , (ak−1 , bk−1 ). Since, by hypothesis, H /ek and (H /ek )/F are L-perfect for any F ⊆ {e1 , . . . , ek−1 }, it follows by the induction hypothesis that Hk−1 /ek is L∗ -perfect. Thus the theorem follows.  The results obtained so far can be applied to build classes of graphs such that the defining linear system of their stable set polytope can be explicitly exhibited. As an example, consider the class P of (P5 , gem)-free graphs defined in [5], i.e., graphs without P5 ’s and gems as induced subgraphs (a P5 is an induced chordless path with 5 nodes and a gem is the graph consisting of a node completely adjacent to an induced chordless path with 4 nodes). 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 find the maximum weight stable set in (P5 , gem)-free graphs [4] and the complete linear description of their stable set polytope is known [15]. Using the terminology introduced in [34], the graphs in P are rank-perfect (equivalently, R-perfect), i.e., the stable set polytope of any graph in P is described only by rank inequalities. Starting from the class P , it is possible to build a much larger class of R-perfect graphs. Note that, besides line graphs [16] and bipartite graphs, a few classes of graphs are known to be rank-perfect, such as antiwebs [35] or some special subclasses of complements of quasi-line graphs [36]. Lemma 28. Let H ∈ P and denote by ΓH a set of pairwise nonincident simplicial edges ab ∈ E (H ) such that

(∗∗) no node in N (a) \ {b} is adjacent to a node in N (b) \ {a}. Then H /F belongs to P for each F ⊆ ΓH . Proof. As the simplicial edges in ΓH form a matching, it is sufficient to prove that, if f = uv is a simplicial edge of H ∈ P that satisfies (∗∗), then H /f ∈ P . Indeed, it is not difficult to see that, if H is P5 -free, then H /f is also P5 -free for any simplicial edge f . Suppose, by contradiction, that H /f contains a gem where q is the node adjacent to the chordless path (p1 , p2 , p3 , p4 ). As H is gem-free, then z ∈ {q, p1 , p2 , p3 , p4 }, where z results from the contraction of f . Since f has the property (∗∗), N (z ) does not contain a chordless path of three nodes. Then z ̸∈ {q, p2 , p3 } and, without loss of generality, assume z = p1 . As (∗∗) holds, either p2 , q ∈ N (a) or p2 , q ∈ N (b). Both cases imply that H contains the gem where q is the node adjacent to the chordless path (t , p2 , p3 , p4 ) with t ∈ {a, b}. A contradiction.  Theorem 29. Let P be the class of (P5 , gem)-free graphs. Then the class G(P , P ) is R-perfect provided that the 2-clique-bond compositions are performed along pairs of nodes corresponding to pairwise nonincident simplicial edges satisfying (∗∗). Proof. Let Hi ∈ G(P , P ) be obtained after i 2-clique-bond compositions according to Definition 22. Observe that a simplicial edge satisfying property (∗∗) is always strongly simplicial. Hence, by Proposition 23, each edge aj bj ∈ ΓHi = {aj bj ∈ ΓH |j > i} is strongly simplicial in Hi . Because of Claim 1 in Proposition 23 and property (∗∗), at most one between aj and bj is adjacent to ai or bi . Since the i-th 2-clique-bond composition introduces edges only between nodes in Hi−1 and nodes in Ti , it follows that these edges have at most one endnode in NHi (aj ) ∪ NHi (bj ), thus implying that aj bj satisfies (∗∗) in Hi . The class P is proved to be R-perfect in [15]. By Lemma 28, if H ∈ P and ΓH is a set of simplicial edges of H that are pairwise nonincident and satisfy property (∗∗), then H /F ∈ P (and so H /F is R-perfect) for any F ⊆ ΓH . Thus, by Theorem 27, the thesis follows.  The class G(P , P ) provides an example on how to use the 2-clique-bond composition to build classes of graphs which are rank-perfect. Note that graphs in G(P , P ) generalize (P5 , gem)-free graphs since they may contain induced P5 ’s. Another interesting family of graphs whose stable set polytope can be investigated using the polyhedral properties of the 2-clique-bond composition is that of claw-free graphs. In fact, Chudnovsky and Seymour [10] have recently decomposed the claw-free graphs using two operations: the 1-join and the generalized 2-join. Given a graph G = (V , E ), a partition (X1 , X2 ) of V is a 1-join if, for i = 1, 2, there exists a nonempty clique Ai ⊂ Xi such that every node in A1 is adjacent to every node in A2 and there are no other edges between X1 and X2 . (Note that this definition of 1-join given in [10] differs from the original one where the sets Ai are not required to be cliques [31].) Clearly, if G admits a 1-join then G has a clique cutset and so, by Theorem 4, a linear description of STAB(G) is obtained as the union of the linear systems describing STAB(Gi ) where Gi is the subgraph of G induced by Xi , i = 1, 2.

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The generalized 2-join is defined in [10] as follows: Suppose that X0 , X1 , and X2 are a partition of V and, for i = 1, 2, there are subsets Yi , Zi of Xi satisfying the following:

• X0 ∪ Y1 ∪ Y2 and X0 ∪ Z1 ∪ Z2 are cliques, and no node of X0 is adjacent to Xi \ (Yi ∪ Zi ) for i = 1, 2, • for i = 1, 2, Yi ∩ Zi = ∅ and Yi , Zi and Xi \ (Yi ∪ Zi ) are all nonempty, • for all v1 ∈ X1 and v2 ∈ X2 , either v1 is not adjacent to v2 , or v1 ∈ Y1 and v2 ∈ Y2 , or v1 ∈ Z1 and v2 ∈ Z2 . The triple (X0 , X1 , X2 ) is called a generalized 2-join. It is straightforward to see that the generalized 2-join is a special case of the 2-clique-bond where A1 = Y1 ∪ X0 , B1 = Z1 ∪ X0 , A2 = Y2 , B2 = Z2 . It follows that every graph that admits a generalized 2-join is obtained by performing a 2-clique-bond composition along pairs of nodes corresponding to simplicial edges ai bi that satisfy the further property N (Ai ∩ Bi ) = N (ai ) ∪ N (bi ) for i = 1, 2. Thus, it is possible to describe the stable set polytope of claw-free graphs starting from the linear description of the stable set polytopes of the building blocks of the decomposition identified in [10]. What is interesting in this case is that the family of inequalities L used to prove L-perfection contains inequalities that are not rank. This will be the topic of two subsequent papers [19,20]. As a final remark, observe that the graph G in Fig. 2 provides an example of graphs that admit a 2-clique-bond but not a generalized 2-join. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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