The stable set polytope of claw-free graphs with stability number at least four. II. Striped graphs are G -perfect

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Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb

The stable set polytope of claw-free graphs with stability number at least four. II. Striped graphs are G-perfect A. Galluccio, C. Gentile, P. Ventura Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, Consiglio Nazionale delle Ricerche (IASI-CNR), viale Manzoni 30, 00185 Roma, Italy

a r t i c l e

i n f o

Article history: Received 15 July 2010 Available online xxxx Keywords: Stable set polytope Claw-free graphs Homogeneous set

a b s t r a c t In [6], Edmonds provided the first complete description of the polyhedron associated with a combinatorial optimization problem: the matching polytope. As the matching problem is equivalent to the stable set problem on line graphs, many researchers tried to generalize Edmonds’ result by considering the stable set problem on a superclass of line graphs: the clawfree graphs. However, as testified also by Grötschel, Lovász, and Schrijver [14], “in spite of considerable efforts, no decent system of inequalities describing STAB(G) for claw-free graphs is known”. Here, we provide an explicit linear description of the stable set polytope of claw-free graphs with stability number at least four and with no 1-join. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Given a graph G = (V, E) and a vector w ∈ QV+ of node weights, the stable set problem is the problem of finding a set of pairwise nonadjacent nodes (stable set) of E-mail addresses: [email protected] (A. Galluccio), [email protected] (C. Gentile), [email protected] (P. Ventura). http://dx.doi.org/10.1016/j.jctb.2014.02.009 0095-8956/© 2014 Elsevier Inc. All rights reserved.

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maximum weight. Let α(G, w) denote the maximum weight of a stable set of G; we refer to α(G) = α(G, 1) (1 being the vector of all ones) as the stability number of G. The stable set polytope, denoted by STAB(G), is the convex hull of the incidence vectors of the stable sets of G. A linear system Ax  b is said to be defining for STAB(G) if STAB(G) = {x ∈ RV : Ax  b}. Chudnovsky and Seymour [4] proved that every claw-free graph that does not admit a 1-join satisfies (at least) one of the following conditions: (i) it has stability number at most 3, or (ii) it is a fuzzy circular interval graph, or (iii) it can be obtained by “properly composing” five types of graphs, called strips: fuzzy linear interval strips (also called Z1 -strips), fuzzy Z2 -strips, fuzzy Z3 -strips, fuzzy Z4 -strips, and fuzzy Z5 -strips. Let Q denote the set of quasi-line graphs that are composition of fuzzy linear interval strips and Qc denote the set of quasi-line graphs that are fuzzy circular interval. We call striped graphs and denote them by C s the claw-free graphs that are obtained by composing: fuzzy linear interval strips, fuzzy Z2 -strips, fuzzy Z3 -strips, fuzzy Z4 -strips, and fuzzy Z5 -strips. Thus the Chudnovsky–Seymour decomposition states that every claw-free graph with stability number at least 4 and without 1-joins belongs either to Qc or to C s . A linear system describing STAB(G) was given by Chudnovsky and Seymour [3] when G ∈ Q and by Eisenbrand et al. [7] when G ∈ Qc . The latter result shows that the rank inequalities are not sufficient to describe STAB(G) as long as G is not a line graph and exhibits a family of non-rank inequalities needed to describe STAB(G), the clique-family inequalities [15], that may be quite complicated even in the case of quasi-line graphs. In a companion paper [12] we studied striped graphs that do not contain Z5 -strips as an induced subgraph. We called these graphs fuzzy antihat graphs and we proved that their stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. Unfortunately, these inequalities are no longer sufficient to describe STAB(G) if G contains a Z5 -strip. In [8,9], the authors were the first to present a new family of inequalities with {0, 1, 2}-coefficients that are facet defining for the stable set polytope of claw-free graphs. These new inequalities are called multiple geared rank inequalities. In this paper we prove that the sequential liftings of multiple geared rank inequalities together with nonnegativity, rank, and lifted 5-wheel inequalities are sufficient to describe STAB(G) when G is a striped graph. This result joined with those in [12] provides the complete linear description of the stable set polytope of claw-free graphs with stability number at least 4 and with no 1-join that are not fuzzy circular interval. It also settles a conjecture presented in [9] and fills the discrepancy between the optimization and the separation problem over stable set polyhedra of claw-free graphs with stability number at least 4 and with no 1-join (see [14,17] for references on this problem). In Section 2 we briefly recall the basic definitions and the polyhedral properties of the 2-clique-bond composition introduced in [11]. In Section 3 we provide the list of

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inequalities that are facet defining for the stable set polytope of a closed fuzzy Z5 -strip G and the graph obtained from G by contracting a prescribed simplicial edge. In Section 4 we summarize the basic results on multiple geared rank inequalities and finally, in Section 5, we present the linear system defining STAB(G) when G is a striped graph. 2. Basic definitions and preliminary results Let G = (V, E) be a simple, connected graph with node set V and edge set E (when we deal with more then one graph we also denote them by V (G) and E(G) to avoid confusion). Two nodes u and v are adjacent (nonadjacent) if uv ∈ E (uv ∈ / E). We denote by N (v) the set of nodes in V adjacent to v and by N [v] the closed neighbourhood of v, namely the set N (v) ∪ {v}. Two adjacent nodes u and v are twins if N [u] = N [v]. It is known that twins have always the same coefficient in every facet inducing inequality for  STAB(G) [5]. Given A ⊂ V , N [A] = v∈A N [v]. We also denote by G \ A the subgraph of G induced by V \ A, where A ⊆ V , and by G + e (G/e) the subgraph of G obtained by adding (contracting) the edge e. Given two subsets of nodes U, Z ⊂ V , we say that U is Z-complete (Z-anticomplete) if every node u ∈ U is adjacent (not adjacent) with every node z ∈ Z. Obviously, U is Z-complete (Z-anticomplete) if and only if Z is U -complete (U -anticomplete). A k-path is a chordless path with k nodes and it is denoted by Pk . A k-hole Ck = (v1 , v2 , . . . , vk ) is a chordless cycle of length k; a k-antihole C k is the complement of a k-hole. A k-wheel (k-antiwheel) W = (h : Ck ) (W = (h : C k )) is a graph consisting of a k-hole Ck (k-antihole C k ) and a node h (hub of W ) adjacent to every node of Ck (C k ). If k = 3, the 3-antiwheel is called claw and denoted by (y : w1 , w2 , w3 ), where y is the center of the claw. If k = 5, then C 5 is isomorphic to C5 and we always refer to W as a 5-wheel. Most of the proofs in this paper use polyhedral techniques from combinatorial optimization and (integer) linear programming that we summarize in the following. Given a vector β ∈ Rm and a subset S ⊆ {1, . . . , m}, define βS ∈ R|S| as the sub vector of β restricted on the indexes of S and let β(S) = i∈S 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 ). 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. 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 subgraph Gβ of G induced by the nonzero coefficients of a valid inequality (β, β0 ) of STAB(G) is called the supporting graph of (β, β0 ). It is not difficult to see that if (β, β0 ) is facet defining for STAB(G), then it is facet defining for STAB(Gβ ). A linear system describing STAB(G) is minimal if it has the minimum number of inequalities. It is a well known result in combinatorial optimization that every inequality in a minimal defining linear system for STAB(G) is facet defining for STAB(G) [17].

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 A linear inequality j∈V πj xj  π0 is said to be a rank inequality for STAB(G) if there exists U ⊆ V such that πi = 1 for each i ∈ U ⊆ V , πi = 0 for each i ∈ V \ U , and π0 = α(G[U ]) where G[U ] is the subgraph of G induced by U . Given a 5-wheel 5 W = (h : v1 , v2 , v3 , v4 , v5 ), the inequality i=1 xvi + 2xh  2 is called 5-wheel inequality and it is known to be facet defining for STAB(W ). We now recall the sequential lifting procedure defined in [16]. Let S (G) denote the  set of 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])

β(S)

(1)

is facet defining for STAB(G). This inequality is called sequential lifting (or simply lifting) of (βV \{v} , 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. Sometimes we refer to lifted inequalities that are obtained by iterating the lifting operation starting from a particular class of inequalities. For instance we refer to “lifted 5-wheel inequalities” as those inequalities obtained by possibly applying a finite sequence of lifting operations starting from 5-wheel inequalities. A node is simplicial if its neighbourhood induces a clique. Given an edge ab, let A = N (a) \ {b} and B = N (b) \ {a}. An edge ab is simplicial if A and B are both cliques. An edge ab is super simplicial if it is simplicial and N [A ∩ B] = N [a] ∪ N [b]. Clearly, adding an edge between two simplicial nodes of a graph produces a simplicial edge. Simplicial edges have an interesting polyhedral feature: Proposition 1. (Galluccio et al. [9]) 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 . To avoid new terminology we use the concepts introduced by Chudnovsky and Seymour [4] as much as possible and we point out where our definitions differ. Definition 2. A strip (G, a, b) consists of a claw-free graph G together with two designated simplicial nodes a, b called the ends of the strip. A closed strip (G, ab) is a strip whose ends are joined by an edge. Definition 3. A homogeneous pair of cliques in G is a pair (A, B) such that: • A and B are cliques in G and A ∩ B = ∅, • |A|  2 or |B|  2, • no node of G \ (A ∪ B) has both a neighbour and a non-neighbour in A, and the same in B.

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In order to take into account homogeneous pairs of cliques in claw-free graphs, Chudnovsky and Seymour [4] introduced the following operation that they named thickening: substitute two specified nodes u and v with two disjoint cliques Xu and Xv so that Xu is neither complete nor anticomplete to Xv . It is not difficult to see that the thickening produces homogeneous pairs of cliques, denoted by (Xu , Xv ), and in the following we will use the adjective fuzzy to indicate the presence of a homogeneous pair of cliques in a strip. For details on all these definitions see [4,12]. The next result proved by Eisenbrand et al. [7] says that we can restrict our attention to homogeneous pair of cliques that are C4 -free. Lemma 4. Let (β, β0 ) be a facet defining inequality of STAB(G). Then there exists a graph G , obtained from G by removing some edges, such that (β, β0 ) is also facet defining for STAB(G ) and no homogeneous pair of cliques of G contains an induced C4 . In [12] we showed that, after eliminating twins and C4 ’s, a few canonical homogeneous pairs of cliques can appear in the supporting graph of a facet defining inequality. They are specified in the following lemma. Lemma 5. Let (β, β0 ) be a facet defining inequality such that Gβ contains a homogeneous pair of cliques (Xu , Xv ) with |Xu | = p and |Xv | = q, p  q  1. If (Xu , Xv ) contains no twins and no induced C4 , then p ∈ {q, q + 1} and the nodes in Xu ∪ Xv can be ordered so that: 1. if p = q then either N (ui ) ∩ Xv = {v1 , v2 , . . . , vq−i+1 } for i = 1, . . . , p, and N (vi ) ∩ Xu = {u1 , u2 , . . . , up−i+1 } for i = 1, . . . , q, or N (ui ) ∩ Xv = {v1 , v2 , . . . , vq−i } for i = 1, . . . , p, and N (vi ) ∩ Xu = {u1 , u2 , . . . , up−i } for i = 1, . . . , q; 2. if p = q +1 then N (ui )∩Xv = {v1 , v2 , . . . , vq−i+1 } for i = 1, . . . , p, and N (vi )∩Xu = {u1 , u2 , . . . , up−i } for i = 1, . . . , q. If (Xu , Xv ) is a canonical homogeneous pair of cliques where |Xu | = p and |Xv | = q, / E and n = p + 1 if up v1 ∈ E. Note with p  q, we define its order as n = p if up v1 ∈ / E and n = q + 1 if that the order may also be symmetrically defined as n = q if u1 vq ∈ u1 vq ∈ E obtaining the same value. Moreover, observe that N (ui ) ∩ Xv = {v1 , . . . , vn−i } and N (vi ) ∩ Xu = {u1 , . . . , un−i } in both case 1 and case 2 of Lemma 5. In Fig. 1 we depict three different canonical homogeneous pairs of cliques for n = 4 where, for each node ui ∈ Xu , the subsets N (ui ) ∩ Xv are the same in the three cases and the same holds for each node vi ∈ Xv . Furthermore, as N [u] ⊆ N [v] implies that βu  βv in any facet defining inequality (β, β0 ) that is not a nonnegativity inequality (see [12]), the following holds:

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Fig. 1. Canonical homogeneous pair of cliques with order n = 4.

Observation 6. Let (β, β0 ) be a facet defining inequality such that Gβ contains a canonical homogeneous pair of cliques (Xu , Xv ) with |Xu | = p and |Xv | = q; then βui  βuj for any 1  i  j  p and βvi  βvj for any 1  i  j  q. Finally, we consider two graph compositions and their polyhedral counterparts. A clique-cutset of G is a complete subgraph whose removal disconnects G. A well known result of Chvátal [5] characterizes the stable set polytope of graphs composed via cliquecutsets. Theorem 7. 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 theorem implies the following: Corollary 8. (Galluccio et al. [12]) Let G be a graph and let (β, β0 ) be a facet defining inequality for STAB(G). Then i) Gβ does not contain a clique-cutset; ii) if u ∈ V (Gβ ) is a simplicial node in Gβ , then Gβ is a maximal clique of G. A graph G = (V, E) admits a 1-join if V can be partitioned into two sets V1 and V2 and, for i = 1, 2, there are subsets Ai of Vi such that: A1 ∪ A2 is a clique, V1 \ A1 and V2 \ A2 are nonempty, and the only edges between V1 and V2 are those between A1 and A2 . Clearly, if G admits a 1-join, then A1 ∪ A2 is a clique-cutset of G. Thus, by Theorem 7, it is not restrictive to consider claw-free graphs that do not admit 1-joins. Suppose now that V0 , V1 and V2 are disjoint subsets with union V (G), and for i = 1, 2 there are subsets Ai , Bi of Vi satisfying the following: • V0 ∪A1 ∪A2 and V0 ∪B1 ∪B2 are cliques, and no node of V0 is adjacent to Vi \(Ai ∪Bi ) for i = 1, 2,

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Fig. 2. The graph G is obtained as the 2-clique-bond composition of G1 and G2 along the pairs (a10 , b10 ) and (a20 , b20 )

• for i = 1, 2, Ai ∩ Bi = ∅ and Ai , Bi and Vi \ (Ai ∪ Bi ) are all nonempty, • for all v1 ∈ V1 and v2 ∈ V2 , either v1 is not adjacent to v2 , or v1 ∈ A1 and v2 ∈ A2 , or v1 ∈ B1 and v2 ∈ B2 . The triple (V0 , V1 , V2 ) is called a generalized 2-join in [4]. We introduced another graph composition that is able to build graphs admitting generalized 2-joins [11]. Definition 9. Let G1 and G2 be two disjoint graphs. Let (ai0 , bi0 ) be an ordered pair of nodes such that ai0 bi0 is a simplicial edge of Gi and let Ai = N (ai0 ) \ {bi0 } and Bi = N (bi0 ) \ {ai0 }, i = 1, 2. The 2-clique-bond composition of G1 and G2 along (a10 , b10 ) and (a20 , b20 ) is the graph G obtained by deleting the nodes ai0 and bi0 , 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 (see Fig. 2). If the edges ai0 bi0 are super simplicial, it is not difficult to check that the 2-clique-bond preserves claw-freeness and produces graphs that admit generalized 2-joins where the role of V0 is played by (A1 ∩ B1 ) ∪ (A2 ∩ B2 ). Definition 10. Let G be a graph with a simplicial edge a0 b0 and let A = N (a0 ) \ {b0 } and B = N (b0 ) \ {a0 }. We call even a facet defining inequality of STAB(G) with nonzero coefficients on a0 and b0 that is different from xa0 + xb0  1. Let z0 be the node in V (G/a0 b0 ) obtained by contracting the edge a0 b0 ; we call odd a facet defining inequality of STAB(G/a0 b0 ) with nonzero coefficient on z0 that is different from x(A ∪ {z0 })  1, x(B ∪ {z0 })  1, and xz0  0. Consider the following inequalities obtained as the composition of inequalities of smaller polytopes: Definition 11. Let G be the 2-clique-bond composition of G1 and G2 along (a10 , b10 ) and (a20 , b20 ), respectively. Let z0i be the node resulting from the contraction of ai0 bi0 , i = 1, 2.

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Fig. 3. Even-odd combination of inequalities.

Let β i x  β0i be an even inequality of STAB(Gi ) and let β j x  β0j be an odd inequality of STAB(Gj /aj0 bj0 ) such that βai i = βbii = βzjj = 1, for i, j ∈ {1, 2} and i = j. 0 0 0 An inequality of the form  v∈V



βvi xv +

(Gi \{ai0 ,bi0 })

v∈V

βvj xv  β0i + β0j − 1

(2)

((Gj /aj0 bj0 )\{z0j })

is said to be an even-odd combination of (β i , β0i ) and (β j , β0j ). Notice that the conditions βai i = βbii = βzjj = 1 are not restrictive because, by 0

0

0

Definition 10, βai i , βbii , βzjj > 0 and, by Proposition 1, βai i = βbii , for i, j = 1, 2 and i = j. 0

0

0

0

0

Example 12. Let G be the graph in Fig. 3(c). The following inequality  v∈•

xv + 2



xv  5

(3)

v∈•

is valid for STAB(G) and it is obtained as an even-odd combination of the odd in  equality v∈• xv + 2 v∈• xv  3 of STAB(G1 /e1 ) in Fig. 3(a) and the even inequality  v∈• xv  3 of STAB(G2 ) in Fig. 3(b). 2 The 2-clique-bond composition has a remarkable polyhedral property: Theorem 13. (Galluccio et al. [11]) Let Gi be a graph with a simplicial edge ei = ai0 bi0 , i = 1, 2, and let G be the 2-clique-bond of G1 and G2 . Then STAB(G) is described by the following inequalities: • nonnegativity inequalities; • clique inequalities induced by A1 ∪ A2 and B1 ∪ B2 ; • facet defining inequalities of STAB(Gi ) with zero coefficients on the endnodes of ei for each i = 1, 2;

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• even-odd combinations of facet defining inequalities of STAB(Gi ) and STAB(Gj /ej ) for each i, j = 1, 2 and i = j. The above result shows explicitly how to combine the facet defining inequalities of four polytopes related to G1 and G2 in order to obtain a defining linear system for STAB(G) when G is the 2-clique-bond of G1 and G2 . For instance, by applying Theorem 13 it can be proved that STAB(G) for G in Fig. 2 is described by nonnegativity, rank, lifted 5-wheel inequalities, and the inequality (3). We conclude this section by citing a result on the stable set polytope of a subset of claw-free graphs. Fuzzy linear interval graphs are defined in [3] as follows: Definition 14. A graph G = (V, E) is said to be fuzzy linear interval if: 1. there is a map φ from V to a line L, and 2. there is a family I of intervals of L (none including another) such that no point of L is an end of more than one interval, so that 3. for u, v ∈ V , if uv ∈ E then {φ(u), φ(v)} is a subset of one interval of I, and if uv ∈ / E then φ(u) and φ(v) are both ends of any interval of I containing both of them (and in particular, if φ(u) = φ(v) then u and v are adjacent). Moreover, if [a, b] is an interval of I such that φ−1 (a) and φ−1 (b) are both nonempty subsets of V and at least one of the sets φ−1 (a) and φ−1 (b) has more than one member, then the interval [a, b] is said to be fuzzy. A fuzzy linear interval graph with two nonadjacent simplicial nodes a0 and b0 is a fuzzy linear interval strip (G, a0 , b0 ) and graphs that are strip composition of fuzzy linear interval strips are called fuzzy line graphs (see also [12]). A linear description of the stable set polytope of fuzzy line graphs is due to Chudnovsky and Seymour (see [18] for a proof): Theorem 15. (Chudnovsky and Seymour [3]) If G is a fuzzy line graph, then STAB(G) is described by nonnegativity and rank inequalities. 3. Fuzzy Z5 -strips In this section we provide a linear description of the stable set polytope of two types of graphs: closed fuzzy Z5 -strips (G, a0 b0 ) and graphs G/a0 b0 obtained by contracting the simplicial edge a0 b0 of G into a single node z0 . We start by giving the original definition of Z5 -strip due to Chudnovsky and Seymour [4]: Definition 16. Let Γ be a graph with node set {d1 , b1 , c, b2 , d2 , a, h1 , h2 , u11 , u12 , u13 , a0 , b0 } and with adjacency as follows. (d1 , b1 , c, b2 , d2 , a) is a hole of Γ of length 6. Next,

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Fig. 4. A closed Z5 -strip.

a0 is adjacent to d1 , b1 ; b0 is adjacent to b2 , d2 ; h1 is adjacent to a, d1 , b1 , c; h2 is adjacent to c, b2 , d2 , a, h1 ; u11 is adjacent to d1 , a, c, b2 , h1 , h2 ; u12 is adjacent to b1 , c, a, d2 , h1 , h2 ; u13 is adjacent to b1 , d1 , b2 , d2 , a0 , b0 . Let U ⊆ {u11 , u12 , u13 }; then the strip (Γ \U, a0 , b0 ) is called a Z5 -strip. A closed Z5 -strip (G, a0 b0 ) is the graph whose node set is V (Γ \U ) and E(G) = E(Γ \ U ) ∪ {a0 b0 }. A closed Z5 -strip with U = {u13 } is depicted in Fig. 4. A contracted closed Z5 -strip is the graph G/a0 b0 obtained from a closed Z5 -strip (G, a0 b0 ) by contracting the edge a0 b0 into the single node z0 . In this section we mainly deal with closed Z5 -strips such that u13 ∈ U that, for short, we call Z5 -strips. We start by providing a linear description of the stable set polytope of a closed Z5 -strip and a contracted closed Z5 -strip with U = ∅. These two graphs have a fixed number of nodes, therefore it is possible to find the above linear description by using a software code such as PORTA [2] or CDD [1]. The output of these codes shows that (see Appendix A for details): Proposition 17. Let (G, a0 b0 ) be a closed Z5 -strip, then the inequalities of a minimal defining linear system for STAB(G) are nonnegativity, rank, and lifted 5-wheel inequalities. Moreover, lifted 5-wheel inequalities have zero coefficients on a0 and b0 . When running the code on a contracted closed Z5 -strip with U = {u11 , u12 }, the following inequality appears: xd1 + xb1 + xa + xc + xd2 + xb2 + xz0 + 2(xh1 + xh2 )  3. This inequality is facet defining for STAB(G/a0 b0 ) and it was named gear inequality in [10,8]. If G is a Z5 -strip containing any subset of nodes in {u11 , u12 }, the gear inequality has to be lifted to become facet defining for STAB(G/a0 b0 ). It turns out that the lifting coefficients of u11 and u12 are always 1 for all possible lifting orders. Thus, the resulting inequality is unique and it is named (lifted) gear inequality:  v∈V (G/a0 b0 )\{h1 ,h2 }

xv + 2(xh1 + xh2 )  3.

(4)

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By running PORTA or CDD, one can see that lifted gear inequalities are the only inequalities that have to be added to nonnegativity, rank, lifted 5-wheel inequalities, to obtain a complete linear description of STAB(G/a0 b0 ) for any contracted closed Z5 -strip (see again Appendix A for details). Proposition 18. Let G/a0 b0 be a contracted closed Z5 -strip, then the inequalities of a minimal defining linear system for STAB(G/a0 b0 ) are nonnegativity, rank, lifted 5-wheel inequalities, and the lifted gear inequality (4). Moreover, lifted 5-wheel inequalities have zero coefficients on the node z0 . Chudnovsky and Seymour [4] showed that homogeneous pairs of cliques in Z5 -strips appear only by thickening the pair of nodes {h1 , h2 }. So, in the following, we simply call fuzzy those Z5 -strips having a homogeneous pair (Xh1 , Xh2 ). First we consider fuzzy closed Z5 -strips, and then we extend our results to complete fuzzy closed Z5 -strips. Since we are mainly interested in characterizing the facet defining inequalities of STAB(G), it is convenient to further specify the structure of the supporting graphs of facet defining inequalities in claw-free graphs. Therefore, because of Lemma 5 and Corollary 8, in the remaining of the paper we shall deal with facet defining inequalities of STAB(G) that are minimal in the following sense: Definition 19. Let (β, β0 ) be facet defining for STAB(G) and let Gβ be its supporting graph. Then we say that (β, β0 ) is minimal if Gβ has no twins and every homogeneous pair of cliques of Gβ is canonical. Thus in the following we denote by hi1 for i = 1, . . . , p the nodes in Xh1 and by hj2 for j = 1, . . . , q the nodes in Xh2 . Proposition 20. Let G/a0 b0 be a contracted fuzzy closed Z5 -strip with a canonical homogeneous pair (Xh1 , Xh2 ) of order n  2. Then, for any maximal clique K ⊆ Xh1 ∪ Xh2 with K ∩ Xh1 = ∅ and K ∩ Xh2 = ∅, the inequality  v∈V 

xv + 2



xv  3,

(5)

v∈K

where V  = V (G/a0 b0 ) \ (Xh1 ∪ Xh2 ), is a lifted gear inequality that is facet defining for STAB(G/a0 b0 ). Proof. Because of Lemma 5 and Observation 6, we may assume that K = {h11 , . . . , hi1 } ∪ }, 1  i < n. Consider the lifted gear inequality (4) with h1 = h11 and h2 = {h12 , . . . , hn−i 2 h12 that is facet defining for STAB(G[V  ]) where V  = V (G)\((Xh1 \{h11 })∪(Xh2 \{h12 })). Then lift the nodes in (Xh1 \ {h11 }) ∪ (Xh2 \ {h12 }) in the following order:

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h21 , . . . , hi1 ,

h22 , . . . , hn−i , 2

p hi+1 1 , . . . , h1 ,

hn−i+1 , . . . , hq2 . 2

Let Vj = V  ∪ {h11 , . . . , hj1 } for j = 2, . . . , i; when lifting node hj1 , G[Vj ] \ N [hj1 ] coincides with {b2 , d2 , z0 }. As at most one node in {b2 , d2 , z0 } belongs to a stable set, the lifting coefficient βhj = 3 − 1 = 2. Similarly, βhj = 2, for j = 2, . . . , n − i. If u ∈ Xh1 \ K, then 1 2 there exists a node v ∈ Xh2 ∩ K such that uv ∈ / E. Therefore {v, z0 } is a tight stable set for the lifted gear inequality and consequently βu = 3 − (2 + 1) = 0. Similarly for nodes v ∈ Xh2 \ K. The inequality resulting from the above lifting procedure is facet defining [16] and the thesis follows. 2 Notice that the lifted gear inequality (4) gives rise to several different facet defining inequalities depending on the choice of the maximal clique K. We now provide linear descriptions for the stable set polytopes of fuzzy closed Z5 -strips and their contractions. To this purpose we need some preliminary results: Lemma 21. Let (G, a0 b0 ) be a fuzzy closed Z5 -strip and let (β, β0 ) be a facet defining inequality of STAB(G) such that Gβ does not contain a 5-wheel. Then (β, β0 ) is a rank inequality. The same result holds for G/a0 b0 . Lemma 21 derives from Theorem 15 by observing that every induced subgraph of a (contracted) fuzzy closed Z5 -strip containing no 5-wheel is a composition of fuzzy linear interval strips. In our proofs we often use the next lemma resulting from the full dimensionality of STAB(G). Lemma 22. Let (β, β0 ) be a facet defining inequality for STAB(G). Then, for every valid inequality (γ, γ0 ) that, up to positive multiplications, is not (β, β0 ), there exists a stable set S that is tight for (β, β0 ) and not for (γ, γ0 ) (i.e., β(S) = β0 and γ(S) < γ0 ). The next two theorems provide a linear description of STAB(G) and STAB(G/a0 b0 ) when (G, a0 b0 ) is a closed fuzzy Z5 -strip. The proofs of these theorems work as follows: we are given a list of facet defining inequalities (γ, γ0 ) of STAB(G) and we need to show that they are sufficient to completely describe STAB(G) if G is a closed fuzzy Z5 -strip. The same approach works for contracted closed fuzzy Z5 -strips, but we need a different list of inequalities. To this aim we suppose, by contradiction, that there exists an inequality (β, β0 ) that is facet defining for STAB(G) and is not included into the list. Then we get contradictions in two possible ways: • either we find an inequality (γ, γ0 ) in the original list that has the same tight stable sets of (β, β0 ), thus contradicting Lemma 22, • or we find a stable set S that is tight for (β, β0 ) and not for a (γ, γ0 ) (such an S exists for each (γ, γ0 ) in the original list, by Lemma 22) and we use S to derive conditions on the coefficients βv , v ∈ V (Gβ ), that lead to a contradiction.

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Theorem 23. Let G be a closed fuzzy Z5 -strip with a simplicial edge a0 b0 . Then STAB(G) is described by the following inequalities: • nonnegativity inequalities, • rank inequalities, • lifted 5-wheel inequalities. Proof. Suppose by contradiction that there exists a minimal facet defining inequality (β, β0 ) for STAB(G) that is not in the above list. If Gβ does not contain a homogeneous pair of cliques (Xh1 , Xh2 ), then the thesis follows from Proposition 17. By Definition 19, (Xh1 , Xh2 ) is canonical with Xh1 = {h11 , . . . , hp1 } and Xh2 = {h12 , . . . , hq2 }. By Lemma 21, we can also assume that Gβ contains at least one 5-wheel, so implying that βa , βc , and the pair βh11 , βh12 are all nonzero and h11 h12 ∈ E(Gβ ), i.e., the order of (Xh1 , Xh2 ) is n  2 and max(p, q)  2. Since a0 b0 is simplicial, by Proposition 1, we can distinguish two cases: Case 1: βa0 = βb0 = 0. Since, by Corollary 8, Gβ does not contain clique-cutsets and Gβ contains a 5-wheel, at most one node in {b1 , d1 , b2 , d2 } has zero coefficient. We select the inequality x(V (Gβ ))  3 as (γ, γ0 ) and we seek for a stable set S that is tight for (β, β0 ) and not for (γ, γ0 ). Clearly, S is maximal in Gβ and |S| = 2. First observe that the only maximal stable sets of Gβ of cardinality two are of type: / V (Gβ )), and {d1 , b2 } {hi1 , b0 } for hi1 ∈ Xh1 , {hi2 , a0 } for hi2 ∈ Xh2 , {b1 , d2 } (if u11 ∈ / V (Gβ )). In the following, we prove that all these stable sets are not tight for (if u12 ∈ (β, β0 ). It is easy to see that the stable sets {hi1 , b0 } are not tight for (β, β0 ), because otherwise at least one between {hi1 , a0 , b2 } or {hi1 , a0 , d2 } violates (β, β0 ). Similarly for {hi2 , a0 }. / V (Gβ ) and the maximal stable set {b1 , d2 } is tight for Suppose now that u11 ∈ (β, β0 ). Then βb1 > βu for any u ∈ {a0 , c} ∪ Xh1 (since otherwise {a0 , c}, or {a0 , hi1 } for i = 1, . . . , p could replace b1 in {b1 , d2 }, so violating (β, β0 )). By Lemma 22, there exists a stable set T that is tight for (β, β0 ) and does not intersect the clique {a0 , b1 , d1 }; then b0 ∈ T and T ∩ {u12 , c, hi1 } = ∅ for some hi1 ∈ Xh1 (since otherwise T ∪ {b1 } violates (β, β0 )). Now, if T ∩ {c, hi1 } = ∅, then T \ {c, hi1 } ∪ {b1 } is feasible and violates (β, β0 ), contradiction. On the other hand, if u12 ∈ T , then T = {u12 , b0 } and, consequently, at least one between T ∪ {d1 } and T \ {b0 } ∪ {a0 , b2 } violates (β, β0 ), contradiction. Hence, / V (Gβ ), then {b1 , d2 } is not tight for (β, β0 ), and symmetrically, if u12 ∈ / V (Gβ ), if u11 ∈ {d1 , b2 } is not tight for (β, β0 ). It follows that every tight stable set of (β, β0 ) contains exactly three nodes, i.e., satisfies as an equality the rank inequality x(V (Gβ ))  3, contradicting Lemma 22. Case 2: βa0 = βb0 = 0. Define Y = V (Gβ ) ∩ {u11 , u12 }. Suppose first that βb1 = 0. Then also βd1 = 0, / E(Gβ ), otherwise Gβ would contain the clique-cutset Xh1 ∪ {a, u11 }. Moreover, if hp1 h12 ∈

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it is not difficult to check that every maximal stable set of Gβ has cardinality two. It follows that all the stable sets tights for (β, β0 ) are also tight for the rank inequality x(V (Gβ ))  2, contradicting Lemma 22. As a consequence, h12 is adjacent to all the nodes of Xh1 . Now consider the inequality (γ, 2) obtained from the 5-wheel inequality 2xh12 + x({h11 , c, b2 , d2 , a})  2 by sequentially lifting the nodes of Xh1 \ {h11 }, Xh2 \ {h12 } and Y in this order. The coefficients βu for all these nodes are equal to 1. As there is no node hi2 ∈ Xh2 \ {h12 } that is Xh1 -complete, all maximal stable sets of Gβ are tight for (γ, 2), so contradicting Lemma 22. Then βb1 > 0 and, using symmetric arguments, we can also prove that b2 , d1 , d2 > 0. Observe that b1 d1 and b2 d2 are both simplicial edges of Gβ ; thus, by Proposition 1, βb1 = βd1 and βb2 = βd2 . Moreover, since N [hij ] ⊃ N [bj ] and N [hij ] ⊃ N [dj ], j = 1, 2, it follows that βhij  βbj = βdj for each hij ∈ Xhj and j = 1, 2. Claim 1. βh11 = βhi1 for each i = 2, . . . , p and βh12 = βhj for each j = 2, . . . , q. 2 Consider the lifted 5-wheel inequality (γ 1 , 2) obtained from 2xh12 + x({h11 , c, b2 , d2 , a})  2 by sequentially lifting the nodes of Xh2 \ {h12 }, Y , and Xh1 \ {h11 }, b1 , d1 , in this order. Each node in Xh2 \ {hn2 , h12 } is lifted with coefficient 2 as they are all adjacent to every node of the 5-wheel; if n = q, then hn2 is lifted with coefficient 1 because it is not adjacent to h11 . Then the nodes in Y are lifted with coefficient 1, while the nodes in Xh1 \ {h11 } are lifted with coefficient zero, as they are not adjacent to hn−1 and hn−1 2 2 has already been lifted with coefficient 2. Similarly, b1 and d1 are lifted with coefficient zero. As V (Gβ ) = V (Gγ 1 ), there exists a stable set S that is tight for (β, β0 ) and satisfies γ 1 (S) < 2. Then S is of type: a) {hj1 , u} with hj1 ∈ Xh1 \ {h11 }, u ∈ {b2 , d2 } or u = hn2 (if n = q), b) {hn2 , v} with v ∈ {b1 , d1 } and n = q. Suppose S is of type b) and no tight stable set of type a) exists. In particular, {h21 , hn2 } is not tight and then βb1 = βd1 > βh21 , a contradiction. Thus a tight stable set of type a) must exist and, as βhj  βh21 for any j  2 by 1 Observation 6, we may assume that S = {h21 , u}. Moreover, since S \ {h21 } ∪ {h11 } is also a stable set, then βh21 = βh11 . We now iterate this procedure, using at each iteration i = 2, . . . , p − 1 the inequality (γ i , 2) obtained from 2xh12 + x({h11 , c, b2 , d2 , a})  2 by sequentially lifting p the nodes of {h22 , . . . , hn−i }, Y , {h21 , . . . , hi1 }, {hn−i+1 , . . . , hq2 }, {hi+1 2 2 1 , . . . , h1 }, {b1 , d1 }, n−i 2 i in this order. The nodes u ∈ {h2 , . . . , h2 } receive a coefficient γu = 2, the nodes u ∈ Y ∪ {h21 , . . . , hi1 } ∪ {hn−i+1 , . . . , hq2 } receive a coefficient γui = 1, and the nodes 2 p i+1 u ∈ {h1 , . . . , h1 } ∪ {b1 , d1 } receive a coefficient γui = 0. As for the case i = 1, every tight stable set S for (β, β0 ) missing (γ i , 2) is of the form j i+1 i S = {hi+1 1 , u} with u ∈ {b2 , d2 , h2 } (j  n − i + 1). Since S \ {h1 } ∪ {h1 } is feasible

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then βhi+1 = βhi1 for i = 2, . . . , p − 1. After p − 1 iterations we obtain that βhi1 = βh11 for 1 all i = 2, . . . , p. Symmetric arguments prove that βh12 = βhi2 for each i = 2, . . . , q. (End of Claim 1) By Lemma 22, there exists a stable set S  that is tight for (β, β0 ) and misses the clique {u12 , b1 , c} ∪ Xh1 . Then S  = {d1 , b2 } (if u12 ∈ / Y ) or S  = {hi2 , d1 } for some hi2 ∈ Xh2 . The former case cannot occur because βb1 = βd1 and the stable set {b1 , a, b2 } would violate (β, β0 ), a contradiction. Thus S  = {hi2 , d1 } and βhi2 > βb2 = βd2 . As, by Claim 1, βhi2 = βh12 for each 2  i  q, we have that βhq2 > βd2 . Symmetric arguments show that the stable set S  = {hp1 , d2 } is tight for (β, β0 ). But then, since hp1 hq2 ∈ / E(Gβ ), the stable set S  \ {d2 } ∪ {hq2 } violates (β, β0 ), a contradiction. 2 Theorem 24. Let G be a closed fuzzy Z5 -strip with a simplicial edge a0 b0 . Then STAB(G/a0 b0 ) is described by the following inequalities: • • • •

nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, lifted gear inequalities.

Proof. Suppose by contradiction that there exists a minimal facet defining inequality (β, β0 ) for STAB(G/a0 b0 ) that is not in the above list. Let Y = V (Gβ ) ∩ {u11 , u12 }. If Gβ does not contain a homogeneous pair of cliques (Xh1 , Xh2 ), then the thesis follows from Proposition 18. By Definition 19, (Xh1 , Xh2 ) is canonical with Xh1 = {h11 , . . . , hp1 } and Xh2 = {h12 , . . . , hq2 }. By Lemma 21, we can also assume that Gβ contains at least one 5-wheel, so implying that βa , βc , and the pair βh11 , βh12 are all nonzero and h11 h12 ∈ E(Gβ ), i.e., the order of (Xh1 , Xh2 ) is n  2 and max(p, q)  2. Moreover, assume also z0 ∈ V (Gβ ), since otherwise we are in Case 2 of Theorem 23 and the thesis follows. Since, by Corollary 8, Gβ does not contain clique-cutsets and Gβ contains a 5-wheel, at most one node in {b1 , d1 , b2 , d2 } has coefficient zero. Suppose that βd1 = 0, then z0 b1 is a simplicial edge and, by Proposition 1, βz0 = βb1 . By Lemma 22, there exists a tight stable set T for (β, β0 ) missing the clique {z0 , b2 , d2 }. It is not difficult to see that b1 ∈ T and that T ∩ Xh2 = ∅. So let T = {b1 , h12 }. Then T  = {z0 , h12 } is also tight and hp1 h12 ∈ E(Gβ ) (i.e., n = p + 1) otherwise T  ∪ {hp1 } violates (β, β0 ). Now consider the set V  = {b1 , b2 , d2 , c, u12 , z0 , h12 } ∪ Xh1 and the rank inequality (γ, 2)  v∈V 

xv  2

(6)

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that is valid for STAB(G/a0 b0 ). By Lemma 22, there exists a stable set S that is tight for (β, β0 ) and such that γ(S) < 2. It is not difficult to see that S may be of the types: {hi2 , z0 }, {hi2 , b1 } with i > 1, or {u11 , z0 } if u12 ∈ / Y . But all these cases cannot occur: in the first case S ∪ {hp1 } violates (β, β0 ), in the second case S \ {b1 } ∪ {z0 , hp1 } violates (β, β0 ), and in the last case S \ {z0 } ∪ {b1 , d2 } violates (β, β0 ) (this because βb1 = βz0 and βd2 , βhp1 > 0). As a consequence βd1 > 0 and by symmetry βb1 , βb2 , βd2 > 0. The next three claims provide relations among the coefficients βv , v ∈ V (Gβ ), and properties of the tight stable sets of (β, β0 ). Claim 1. If {b1 , d2 } or {u12 , z0 } is tight, then βb1 , βd2  βz0 and βz0 > βd1 , βb2 . / Y . First suppose that {b1 , d2 } Clearly, if {b1 , d2 } or {u12 , z0 } is tight, then u11 ∈ is tight for (β, β0 ); then βb1 > βd1 and βd2 > βb2 (otherwise {d1 , c, d2 } and {b1 , a, b2 } would violate (β, β0 )). By Lemma 22, there exist tight stable sets Si for (β, β0 ) missing the clique {z0 , bi , di } for i = 1, 2; then S2 b1 (as βb1 > βd1 ) and S1 d2 (as βd2 > βb2 ), implying that βb1  βz0 and βd2  βz0 . Moreover, let S  and S  be two tight stable sets missing the cliques {b1 , c} ∪ Xh1 ∪ Y and {d2 , a} ∪ Xh2 ∪ Y , respectively. If d1 ∈ S  then S  \ {d1 } ∪ {b1 } violates (β, β0 ). Thus z0 ∈ S  otherwise b1 can be simply added to S  , so violating (β, β0 ). Symmetrically, z0 ∈ S  . As a consequence, βz0  βb1 , βd2 . As βb1 > βd1 and βd2 > βb2 , βz0 > βd1 , βb2 , as claimed. Suppose now that {u12 , z0 } is tight for (β, β0 ). Then Y = {u12 } and βz0 > βd1 , βb2 , because {d1 , u12 , b2 } is a stable set. Now let Si be a stable set that is tight for (β, β0 ) and misses the clique {z0 , bi , di } for i = 1, 2. Then S2 b1 and S1 d2 , thus implying that βb1  βz0 and βd2  βz0 , as claimed. (End of Claim 1) Claim 2. The stable sets that are tight for (β, β0 ) and do not intersect Xh1 ∪ Xh2 have cardinality three and they are tight for any lifted gear inequality (5). First observe that the maximal stable sets of Gβ of cardinality two that do not contain nodes in Xh1 ∪ Xh2 are at most four: / V (Gβ ), • {b1 , d2 } and {u12 , z0 } if u11 ∈ • {d1 , b2 } and {u11 , z0 } if u12 ∈ / V (Gβ ). Let us prove that when u11 ∈ / V (Gβ ) the first two stable sets cannot be tight for (β, β0 ). Suppose conversely that {u12 , z0 } or {b1 , d2 } is tight for (β, β0 ). Then, by Claim 1, βb1 , βd2  βz0 and βz0 > βd1 , βb2 . By Lemma 22, there exists a stable set T1 that is tight for (β, β0 ) and misses the clique {b1 , c} ∪ Xh1 ∪ Y . It is not difficult to verify that z0 ∈ T1 , because if d1 ∈ T1 then T1 \ {d1 } ∪ {b1 } violates (β, β0 ) and if T1 ∩ {d1 , z0 } = ∅ then T1 ∪ {b1 } violates (β, β0 ). If a ∈ / T1 and T1 ∩ Xh2 = ∅, then T1 ∪ {h11 } violates (β, β0 ). But if T1 = {a, z0 }, then T1 ∪{c} violates (β, β0 ), a contradiction. Consequently T1 ∩Xh2 = ∅. Thus T1 = {h12 , z0 }, hp1 h12 ∈ E(Gβ ) (that is, n = p + 1) and βz0  βb1 . Symmetrically, there exists a stable

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set T2 = {h11 , z0 } that is tight for (β, β0 ) and misses the clique {d2 , a} ∪ Xh2 ∪ Y , h11 hq2 ∈ E(Gβ ) (that is, n = q + 1 and p = q), and βz0  βd2 . Hence, βz0 = βb1 = βd2 . Consider now the rank inequality (γ, 2) defined in (6); in this case we have that the stable set S that is tight for (β, β0 ) and such that γ(S) < 2 is either S = {b1 , hi2 } with i  2, or S = {d1 , h12 }, or, in case u12 ∈ / V (Gβ ), S = {d1 , b2 }. Since, in the latter two cases, S \ {d1 } ∪ {b1 } violates (β, β0 ) by Claim 1, S = {b1 , hi2 } for i  2 and we may assume i = 2, as βhi2  βhi+1 for i = 1, . . . , q − 1. Since βb1 = βz0 , we conclude that 2 S \ {b1 } ∪ {z0 , hp1 } violates (β, β0 ), a contradiction. Symmetric arguments prove that, if u12 ∈ / V (Gβ ), then {u11 , z0 } and {d1 , b2 } are not tight for (β, β0 ). (End of Claim 2)

Claim 3. βh11 = βhi1 for each i = 2, . . . , p and βh12 = βhj for each j = 2, . . . , q. 2 Recall that βhij  βhi−1 for i = 1, . . . , |Xhj | and j = 1, 2 by Observation 6. Suppose j by contradiction that there exists k ∈ {2, . . . , p} such that βhk1 < βhk−1 . 1 As STAB(G/a0 b0 ) is full dimensional, for each node u ∈ V (G/a0 b0 ) there exists a stable set that is tight for (β, β0 ) and contains u. We prove that the only stable set S that is tight for (β, β0 ) and that contains hk1 is S = {hk1 , hn−k+1 , z0 } where n is the order 2 of (Xh1 , Xh2 ). Suppose conversely that S = {hk1 , u} with u ∈ {b2 , d2 } or S = {hk1 , hj2 , z0 } } violates (β, β0 ), a contradiction. for some j ∈ {n − k + 2, . . . , q}. Then S \ {hk1 } ∪ {hk−1 1 n−k+1 1 Consider now the clique K = {h11 , . . . , hk−1 } ∪ {h , }. By Proposition 20, 2 . . . , h2 1 the following lifted gear inequality, denoted as (γ, 3),      x V Gβ \ (Xh1 ∪ Xh2 ) + 2x h11 , . . . , hk−1 , h12 , . . . , hn−k+1  3, 1 2 is facet defining for STAB(G/a0 b0 ) (note that n − k + 1  q as k  2). By Lemma 22, there exists a tight stable set T for (β, β0 ) such that γ(T ) < 3. By Claim 2 T ∩ (Xh1 ∪ Xh2 ) = ∅. Suppose first that T ∩Xh1 = ∅. For what stated before, the only stable set that contains hk1 and that is tight for (β, β0 ) is S = {hk1 , hn−k+1 , z0 }. Since γ(S) = 3, hk1 ∈ / T . Suppose 2 r s now that h1 ∈ T for some r > k. Then T contains a node h2 , with s  n−k+1. If not, then either T = {hr1 , u} with u ∈ {b2 , d2 } or T = {hr1 , hj2 , z0 } for some j ∈ {n − k + 2, . . . , q}. } violates (β, β0 ), a contradiction. Hence T = {hr1 , hs2 , z0 }, with But then T \ {hr1 } ∪ {hk−1 1 r > k and s  n − k + 1, and γ(T ) = 3, a contradiction. Finally, suppose that hr1 ∈ T for some r < k. Then T also contains a node in V (Gβ ) \ (Xh1 ∪ Xh2 ) and again γ(T ) = 3, a contradiction. It follows that T ∩ Xh1 = ∅. Thus, T = {hn−k+2 , u1 } with u1 ∈ {b1 , d1 } and, consequently, βhn−k+2 = βhn−k+1 . It 2 2

2

follows that S  = S \ {hk1 , hn−k+1 } ∪ {hk−1 , hn−k+2 } violates (β, β0 ), a contradiction. 2 1 2 Symmetrically, it can be proved that βhj = βh12 for each j = 2, . . . , q. 2 (End of Claim 3)

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By Lemma 22, there exists a stable set T1 that is tight for (β, β0 ) and such that T1 ∩ {b1 , d1 , z0 } = ∅. It is not difficult to see that T1 = {hi1 , u2 } with i ∈ {1, . . . , p} and u2 ∈ {b2 , d2 }. By Observation 6 we may assume i = 1. Symmetrically, there exists T2 = {h12 , u1 } with u1 ∈ {b1 , d1 } that is also tight for (β, β0 ). Since hp1 hq2 ∈ / E(Gβ ), it follows that βh11 + βu2  βhp1 + βhq2 + βz0 . By Claim 3, βh11 = βhp1 and so, βu2 > βhq2 = βh12 . But then T2 \ {h12 } ∪ {u2 } is a stable set violating (β, β0 ). A contradiction. 2 To complete this section we extend the above results to Z5 -strips containing u13 . Theorem 25. Let G be a fuzzy closed Z5 -strip with a simplicial edge a0 b0 . Then STAB(G) and STAB(G/a0 b0 ) are described by the following inequalities: • • • •

nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, lifted gear inequalities.

Proof. If G does not contain u13 , the theorem follows from Theorems 23 and 24. So, suppose that u13 ∈ V (G). A fuzzy closed Z5 -strip (G, a0 b0 ) containing u13 is the 2-clique-bond of a fuzzy closed Z5 -strip H = (G \ {u13 }, v1 v2 ) and a 4-wheel W4 = (u13 : u1 , u2 , b0 , a0 ) along the pairs (v1 , v2 ) and (u1 , u2 ), respectively. Thus, by Theorem 13, the facet defining inequalities of STAB(G) are inequalities of types i), ii) and iii) plus: a. even-odd combinations of facet defining inequalities of STAB(H) having nonzero coefficients on v1 and v2 with the unique facet defining inequality of STAB(W4 /u1 u2 ) having nonzero coefficient on the node u0 resulting from the contraction of u1 u2 , that is the clique inequality xa0 + xb0 + xu13 + xu0  1; b. even-odd combinations of facet defining inequalities of STAB(H/v1 v2 ) having nonzero coefficient on the node v0 resulting from the contraction of v1 v2 with the unique facet defining inequality of STAB(W4 ) having nonzero coefficients on u1 and u2 , that is the clique inequality xu1 + xu2 + xu13  1. From Theorems 23 and 24, it follows that the inequalities a) and b) are either rank or lifted gear inequalities. Indeed, lifted gear inequalities of the form shown in Eq. (5) supported by V (G\{a0 , b0 }) and with z0 replaced by u13 , are facet defining for STAB(G) and they are obtained as even-odd combinations of the lifted gear inequalities supported by V (H/v1 v2 ) and the clique inequality xu1 + xu2 + xu13  1. Consider now the contracted fuzzy closed Z5 -strip G/a0 b0 . It is easy to see that u13 is a twin of z0 in G/a0 b0 , and so the linear description of STAB(G/a0 b0 ) is the same as in Theorem 24. 2

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Fig. 5. A graph obtained by two applications of 2-clique-bond composition.

4. Multiple geared rank inequalities In this section we explain the structure of the inequalities obtained by several applications of the 2-clique-bond composition. We start with an example. Consider the graphs in Fig. 2 and rename G2 as H0 , G as H1 and the pair (a20 , b20 ) as (x1 , y1 ). Now, H1 is obtained as the 2-clique-bond composition of H0 and the closed Z5 -strip G1 with U = {u11 , u12 , u13 } along (x1 , y1 ) and (a10 , b10 ) where f1 = x1 y1 ∈ E(H0 ) and e1 = a10 b10 ∈ E(G1 ). By Theorem 13, STAB(H1 ) is described by nonnegativity, rank, 5-wheel inequalities, and by inequalities obtained as even-odd combinations of rank inequalities that are facet defining for STAB(H0 ) and the gear inequality (4) that is facet defining for STAB(G1 /e1 ). The last type of inequalities is depicted in Fig. 3. The inequalities that are even-odd combination of a rank inequality and a gear inequality of the form shown in (4) were called geared rank inequalities in [9]. If the Z5 -strip contains nodes in {u11 , u12 , u13 }, the even-odd combination is performed between a rank inequality and the lifted gear inequality (4) and produces a lifted geared rank inequality. Now perform the 2-clique-bond composition of H1 and the closed Z5 -strip G2 with U = {u11 , u12 , u13 } along (x2 , y2 ) and (a20 , b20 ) where f2 = x2 y2 ∈ E(H1 ) and e2 = a20 b20 ∈ E(G2 ), so to obtain the graph H2 in Fig. 5. Again, by Theorem 13, the stable set polytope of H2 is described by nonnegativity, rank, lifted 5-wheel inequalities, and: 1) even-odd combinations of two rank inequalities, 2) even-odd combinations of a geared rank inequality and a rank inequality, 3) even-odd combinations of geared rank inequality and a gear inequality. The first type of even-odd combination of inequalities produces rank inequalities. The second type produces geared rank inequalities (see Figs. 6, 7, and 8) and the last type produces an inequality that contains two pairs of coefficients 2 that are associated with the hubs of the two closed Z5 -strips (see Fig. 9). Inequalities of types 2) and 3) in the previous example are called multiple geared rank inequalities.

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Fig. 6. Even-odd combination of a geared rank inequality of STAB(H1 /f2 ) and a rank inequality of STAB(G2 ).

Fig. 7. Even-odd combination of a geared rank inequality of STAB(H1 ) and a rank inequality of STAB(G2 /e2 ).

Fig. 8. Even-odd combination of a rank inequality of STAB(H1 ) and the gear inequality of STAB(G2 /e2 ).

Thus, repeated applications of the 2-clique-bond composition with closed Z5 -strips produce multiple geared rank inequalities that contain at least one pair of hubs of Z5 with coefficients 2. Figs. 6, 7, 8, 9 also show that the number of these inequalities is exponential in the number of 2-clique-bond compositions with closed Z5 -strips, because closed Z5 -strips and contracted closed Z5 -strips might both generate inequalities having coefficient 2 on any subset of pairs of hubs.

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Fig. 9. Even-odd combination of a geared rank inequality of STAB(H1 ) and the gear inequality of STAB(G2 /e2 ).

A formal definition of these inequalities is: Definition 26. A valid inequality (γ, γ0 ) belongs to the class GR if and only if (γ, γ0 ) is either a rank inequality, or a lifted gear inequality, or an even-odd combination of two inequalities in GR . The inequalities in GR that are not rank are called lifted multiple geared rank inequalities. Now, let G = (V, E) be a graph, L be a class of inequalities, and consider the polyhedron LSTAB(G) = {x ∈ RV+ | x satisfies L}. We say that a graph G is L-perfect if STAB(G) is described only by inequalities in L, that is if STAB(G) = LSTAB(G). In [9] we considered the following family G of inequalities: Definition 27. A valid inequality (γ, γ0 ) for the stable set polytope belongs to G if and only if (γ, γ0 ) is either a rank inequality, or a lifted 5-wheel inequality, or a lifted gear inequality, or an even-odd combination of two inequalities in G. In the same paper we asked for significant classes of graphs that are G-perfect. In [12] we considered the class W consisting of: rank and lifted 5-wheel inequalities, and we proved that a large subclass of claw-free graphs with no 1-join and with stability number at least four (the so-called fuzzy antihat graphs) are W-perfect. Clearly, W and GR are subclasses of G.

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In the remaining of this paper we consider the subclass of G obtained as the union of GR and W, i.e., WG R = W ∪ GR . The class WG R is a subclass of G since even-odd combinations involving lifted 5-wheel inequalities are present in G but not in WG R . In the next final section we prove that striped graphs are WG R -perfect, thus implying that they are G-perfect. 5. The stable set polytope of striped graphs In this section we present our main result, that is the defining linear system of the stable set polytope of striped graphs. Chudnovsky and Seymour in [4] describe the following construction to build claw-free graphs with α(G)  4 which are not fuzzy circular interval graphs: a) Start with a graph G0 that is a disjoint union of cliques. Take a partition of V (G0 ) into stable sets X1 , . . . , Xk such that each Xi satisfies 1  |Xi |  2. b) For 1  i  k, take a graph Gi (where G0 , G1 , . . . , Gk are pairwise vertex-disjoint) and a subset Yi ⊆ V (Gi ) with |Yi | = |Xi |. When we apply this, the pairs (Gi , Yi ) will be taken from an explicit list of allowed pairs. In particular, Gi is claw-free, Yi is stable in Gi , and for each vertex in Yi , its neighbour set in Gi is a clique. Take the disjoint union of G0 , . . . , Gk . c) For 1  i  k, take a bijection between Xi and Yi ; for each x ∈ Xi and its mate y ∈ Yi , make the neighbour set of x strongly complete to the neighbour set of y, and then delete x, y (the order of these operations does not affect the final outcome). The explicit list mentioned in this construction is called Z0 and consists of 15 members Z1 , . . . , Z15 (see [4] at pp. 1388–1389 for details). In the remaining of this paper we shall refer to this construction as the Chudnovsky–Seymour construction. Since we are interested in claw-free graphs without 1-joins, we take into consideration only those members (Gi , Yi ) of Z0 that give rise to generalized 2-joins in the resulting graph, i.e., those having |Yi | = 2: fuzzy linear interval strips, fuzzy Z2 -strips, fuzzy Z3 -strips, fuzzy Z4 -strips, and fuzzy Z5 -strips (belonging to the classes Z1 , Z2 , Z3 , Z4 , Z5 in [4], respectively). So, the structural theorem for claw-free graphs can be stated as follows: Theorem 28. (Chudnovsky and Seymour [4]) For every claw-free graph G with stability number at least four and with no 1-join, one of the following holds: 1. G can be constructed as above where all pairs (Gi , Yi ) are either fuzzy linear interval strips, fuzzy Z2 -strips, fuzzy Z3 -strips, fuzzy Z4 -strips, fuzzy Z5 -strips, or 3-paths with the internal node substituted by a clique and with endnodes Yi , 2. G is fuzzy circular interval.

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The graphs obtained as in point 1. of Theorem 28 are called striped graphs. Let us denote by G+ the closed strip obtained from the pair (G, Y ), |Y | = 2, by adding an edge e between the two nodes of Y . To avoid further notation, in the following we say that a closed fuzzy strip G+ belongs to Zi if G ∈ Zi , i = 1, . . . , 5. Observation 29. Let K1 and K2 be two disjoint cliques of G0 in the Chudnovsky–Seymour construction and let (y1 , K, y2 ) be a 3-path whose internal node has been substituted by the clique K. Let X1 = {x11 , x21 } and X2 = {x12 , x22 } be two pairs of nodes of G0 such that xi1 , xi2 ∈ Ki , i = 1, 2. Performing the step c) of the Chudnovsky–Seymour construction between X1 and Y = {y1 , y2 }, amounts to remove the nodes {y1 , y2 , x11 , x21 } and set K complete to K1 \ {x11 } and K2 \ {x21 }. In the resulting graph, N (x12 ) ∩ N (x22 ) = K and moreover, N [N (x12 ) ∩ N (x22 )] = N [x12 ] ∪ N [x22 ]. Theorem 28 can be rephrased in terms of 2-clique-bond composition as follows: Theorem 30. Every connected striped graph G can be obtained from a fuzzy line graph H0 by iteratively performing 2-clique-bond compositions of fuzzy closed strips G+ i belonging to Z2 , Z3 , Z4 , and Z5 along a set of endnodes of pairwise non-incident super simplicial edges fi of H0 for i = 1, . . . , n. Proof. Since the order of the compositions in the Chudnovsky–Seymour construction does not affect the final graph G, we may consider the graph obtained from G0 by composing first all fuzzy linear interval strips in Z1 and then all 3-paths. Since 3-paths are linear interval strips (according to Definition 14), the graph H resulting from these compositions is fuzzy line. Let us number from 1 to n the pairs of nodes Xi as well as the pair (Gi , Yi ) that remain to be composed. Then add to H a new edge fi between each pair of simplicial nodes in Xi = {x1i , x2i }, i = 1, . . . , n. The resulting graph H0 is fuzzy line because adding an edge to a fuzzy line graph is the same as composing such a graph with a 4-path (see also Observation 1 in [12]). Moreover, each newly added edge fi = x1i x2i is super simplicial by Observation 29. Finally, the edges fi are pairwise non-incident because the pairs Xi are pairwise disjoint for i = 1, . . . , n. So, let G+ i be the closed strip obtained from the pair (Gi , Yi ) by adding the edge ei between the nodes in Yi , i = 1, . . . , n and let Hi be defined recursively as the 2-clique-bond 1 2 1 2 composition of Hi−1 and G+ i along the pairs of nodes (xi , xi ) and (yi , yi ), i = 1, . . . , n. Note that the edges ft with i  t  n are super simplicial in Hi because the arguments used to prove their super-simpliciality in H0 hold at each iteration. Moreover, the operation described in the Chudnovsky–Seymour construction is equivalent to the 2-clique-bond composition of two graphs along pairs of nodes that are endnodes of super simplicial edges. It follows that the graph G is obtained from H0 by iteratively perform1 2 ing the 2-clique-bond composition of the fuzzy closed strip G+ i along the pairs (xi , xi ) 1 2 and (yi , yi ), i = 1, . . . , n. 2

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24

In the following we show that the 2-clique-bond composition preserves the WG R -perfectness provided that the closed strips belongs to Z2 ∪ Z3 ∪ Z4 ∪ Z5 . Theorem 31. Let G be a graph obtained as the 2-clique-bond of a claw-free graph H and a fuzzy closed strip Z + belonging to Z2 ∪ Z3 ∪ Z4 ∪ Z5 , along pairs (u, v) and (a0 , b0 ) such that f = uv is a super simplicial of H and e = a0 b0 is a super simplicial edge of Z + . If H and H/f are WG R -perfect, then G is WG R -perfect. Proof. Let ze (zf ) denote the node resulting from the contraction of the edge e (f , respectively). By Theorem 13, STAB(G) is described by the following inequalities: i) ii) iii) iv) v) vi)

nonnegativity inequalities; clique inequalities; facet defining inequalities of STAB(H) with zero coefficients on the endnodes of f ; facet defining inequalities of STAB(Z + ) with zero coefficients on the endnodes of e; even-odd combinations of facet defining inequalities of STAB(H) and STAB(Z + /e); even-odd combinations of facet defining inequalities of STAB(H/f ) and STAB(Z + ).

By hypothesis, STAB(H) and STAB(H/f ) are described by nonnegativity, rank, lifted 5-wheel inequalities, and lifted multiple geared rank inequalities. Since f is super simplicial and H is claw-free, it is easy to verify that neither the endnodes of f nor its neighbours can be the hub of a 5-wheel. The same holds for the node zf and its neighbours in H/f (see also Corollary 8 in [12]). It follows that the only inequalities of STAB(H) and STAB(H/f ) having nonzero coefficients on the endnodes of f and zf , respectively, (those involved into even-odd combinations) are rank inequalities or lifted multiple geared rank inequalities. Notice that the coefficients of these inequalities on the endnodes of f and on zf are all ones. If Z + ∈ Zi , for i = 2, 3, 4, then STAB(Z + ) and STAB(Z + /e) are described only by nonnegativity, rank, and lifted 5-wheel inequalities and only the rank inequalities can have nonzero coefficients on the endnodes of e in STAB(Z + ) and on the node ze in STAB(Z + /e) (see Theorems 30, 38, and 41 in [12]). If Z + ∈ Z5 , then, by Theorem 25, STAB(Z + ) and STAB(Z + /e) are described only by nonnegativity, rank, lifted 5-wheel inequalities and lifted gear inequalities (5). Moreover, only rank inequalities and lifted gear inequalities have nonzero coefficients on the endnodes of e in STAB(Z + ) and on the node ze in STAB(Z + /e). Thus, the even-odd combinations v) and vi) resulting from the 2-clique-bond of H and Z + are only rank inequalities or lifted multiple geared rank inequalities, i.e., inequalities in WG R . Thus the theorem follows. 2 We can now state the main result of the paper: Theorem 32. Let G be a striped graph. Then STAB(G) is described by:

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• • • •

25

nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, lifted multiple geared rank inequalities.

Proof. By Theorem 30, G is obtained from a fuzzy line graph H0 by k applications of the 2-clique-bond composition with graphs G+ i ∈ Z2 ∪ Z3 ∪ Z4 ∪ Z5 , i = 1, . . . , k along pairs (x1i , x2i ) and (yi1 , yi2 ) such that: ΓH = {fi = x1i x2i , i = 1, . . . , k} is a set of pairwise non-incident super simplicial edges of H0 and ei = yi1 yi2 is a super simplicial edge of G+ i for i = 1, . . . , k. First observe that the class of fuzzy line graphs is closed under contraction of super simplicial edges; indeed, H/f is obtained by composing the fuzzy line strip (H −f, v1 , v2 ) with the strip (P, a0 , b0 ) where P is the path (a0 , z0 , b0 ). Thus, H/F is fuzzy line for any set F ⊆ ΓH . The rest of the proof is by induction on k. If k = 1 then H1 is WG R -perfect by Theorem 31 and the thesis follows. Now, suppose that the thesis holds up to the (k − 1)-th iteration and let Hk be obtained as the 1 2 1 2 2-clique-bond composition of Hk−1 and G+ k along the pairs (xk , xk ) and (yk , yk ). By Theorem 31, in order to prove that Hk is WG R -perfect it suffices to show that Hk−1 and Hk−1 /fk are WG R -perfect. Since Hk−1 is WG R -perfect by inductive hypothesis, it remains to show that Hk−1 /fk is WG R -perfect. As the edges {f1 , f2 , . . . , fk } are pairwise non incident, the edges {f1 , f2 , . . . , fk−1 } are super simplicial in H/fk . This implies that the graph Hk−1 /fk can be obtained from + + the graph H/fk by k − 1 iterated 2-clique-bond compositions of G+ 1 , G2 , . . . , Gk−1 along the pairs (x11 , x21 ), . . . , (x1k−1 , x2k−1 ). Since H/fk is fuzzy line, it follows that Hk−1 /fk is WG R -perfect and the theorem follows. 2 The above theorem together with the characterization of rank facet defining inequalities in [13] provides the minimal defining linear system for the stable set polytope of striped graphs. Moreover, Theorem 32 together with the result of Eisenbrand et al. [7] on fuzzy circular interval graphs and Theorem 28 yields the following: Theorem 33. Let G be a claw-free graph that has stability number at least four and that does not admit a 1-join. Then STAB(G) is described by: • • • • •

nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, lifted multiple geared rank inequalities, clique-family inequalities.

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Acknowledgment We are deeply indebted to an anonymous referee whose valuable suggestions allowed us to considerably improve the quality of this paper. Appendix A In this appendix we show the outputs of the code PORTA [2] that we used to produce the description of the stable set polytope of a closed Z5 -strips. When the input graph is a closed Z5 -strip that contains {u11 , u12 }, i.e., the graph depicted in Fig. 4, the output of PORTA (excluding nonnegativity inequalities) is the following: c1:

+ xd1 + xh1 + xa

c2:

+ xh1 + xa

c3:

+ xh1

c4: c5:

+xb1 + xd1 + xh1

c6:

+xb1 + xd1

+ xu11

1

+ xh2

+ xu11

1

+ xc + xh2

+ xu11

1

+ xc + xh2 + xb2

+ xu11

1 1 + xa0

c7:

+ xh1 + xa

+ xh2

c8:

+ xa

+ xh2

c9:

+xb1

c10:

+ xd2

1

+ xu12

1

+ xu12

1

+ xh1

+ xc

+ xu12

1

+ xh1

+ xc + xh2

+ xu12

1 1

+ xh2 + xb2 + xd2

c11:

+ xa0 + xb0  1

c12:

+ xb0  1

+ xb2 + xd2

c13: c14:

+ xd1

c15:

+ xd1

+ xb2 + xa

+ xa0 + xb0  2

+ xu11 + xd2

+ xa0 + xb0  2

+ xd2

+ xu12 + xa0 + xb0  2

c16:

+xb1

c17:

+xb1

c18:

+xb1 + xd1 + xh1 + xa + xc + xh2 + xb2 + xd2 + xu11 + xu12 + xa0 + xb0  3

c19:

+xb1 + xd1 + 2xh1 + xa + xc + xh2

c20:

+ xc

+ xa0 + xb0  2

+ xb2 + xu11 + xu12

2

+ xh1 + xa + xc + 2xh2 + xb2 + xd2 + xu11 + xu12

2

The inequalities from c1 to c18 are rank inequalities. The inequalities c19 and c20 are lifted 5-wheel inequalities with coefficient zero on a0 and b0 . This list proves Proposition 17. When the input is a contracted closed Z5 -strip that contains {u11 , u12 }, the output of PORTA (excluding nonnegativity inequalities) is the following:

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c1:

+ xh1 + xa

c2:

+ xd1 + xh1 + xa

c3:

+ xh1

c4:

+ xu11

1

+ xu11

1

+ xc + xh2

+ xu11

1

+ xc + xh2 + xb2

+ xu11

1

+ xh2

c5:

+ xh1 + xa

+ xh2

c6:

+ xa

+ xh2

c7:

+ xd2

+ xu12

1

+ xu12

1

+ xh1

+ xc + xh2

+ xu12

1

+ xh1

+ xc

+ xu12

1

c8:

+xb1

c9:

+xb1 + xd1 + xh1

1

c10:

+ xh2 + xb2 + xd2

1

c11:

+ xb2 + xd2

+ xz0  1 + xz0  1

c12: +xb1 + xd1 c13:

+ xd1 + xh1 + xa

c14: +xb1 + xd1 + xh1

+ xz0  2

+ xh2 + xb2 + xd2 + xu11 + xc + xh2 + xb2

+ xz0  2

+ xu11 + xd2

+ xu12 + xz0  2

+ xh1

+ xc + xh2 + xb2 + xd2

+ xu12 + xz0  2

c17: +xb1 + xd1 + xh1

+ xh2 + xb2 + xd2

+ xz0  2

c15: +xb1 + xd1 + xh1 + xa c16: +xb1

+ xh2

+ xu11 + xu12

2

+ xh1 + xa + xc + 2xh2 + xb2 + xd2 + xu11 + xu12

2

c18: +xb1 + xd1 + 2xh1 + xa + xc + xh2 c19:

27

c20: +xb1 + xd1 + 2xh1 + xa + xc + 2xh2 + xb2 + xd2 + xu11 + xu12 + xz0  3 The inequalities from c1 to c17 are rank inequalities. The inequalities c18 and c19 are lifted 5-wheel inequalities with coefficient zero on z0 . The inequality c20 is the lifted gear inequality (4). This list proves Proposition 18. References [1] D. Avis, K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discrete Comput. Geom. 8 (1992) 295–313. [2] T. Christof, A. Löbel, PORTA – polyhedron representation transformation algorithm, 2004. [3] M. Chudnovsky, P. Seymour, The structure of claw-free graphs, in: Surveys in Combinatorics, in: London Math. Soc. Lecture Note Ser., vol. 327, 2005. [4] M. Chudnovsky, P. Seymour, Claw-free graphs V: Global structure, J. Combin. Theory Ser. B 98 (2008) 1373–1410. [5] V. Chvátal, On certain polytopes associated with graphs, J. Combin. Theory Ser. B 18 (1975) 138–154. [6] J. Edmonds, Maximum matching and a polyhedron with 0, 1 vertices, J. Res. Natl. Bur. Stand. B 69B (1965) 125–130. [7] F. Eisenbrand, G. Oriolo, G. Stauffer, P. Ventura, The stable set polytope of quasi-line graphs, Combinatorica 28 (1) (2008) 45–67. [8] A. Galluccio, C. Gentile, P. Ventura, Gear composition and the stable set polytope, Oper. Res. Lett. 36 (2008) 419–423. [9] A. Galluccio, C. Gentile, P. Ventura, Gear composition of stable set polytopes and G-perfection, Math. Oper. Res. 34 (2009) 813–836.

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