Gear composition and the stable set polytope

Galluccio, Anna; Gentile, Claudio; Ventura, Paolo

doi:10.1016/j.orl.2008.01.003

Cited by 16 publications

(11 citation statements)

References 16 publications

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“…The clique inequality x 4 + x 5 + 4, 5}. Applying Theorem 3.4, we obtain the facet-defining inequality 2x 1 + x 2 + 2x 3 + 2x 4 + x 5 + x 6 + x 7 + x 8 ≤ 3.…”

Section: Theorem 34 Supposementioning

confidence: 91%

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A New Facet Generating Procedure for the Stable Set Polytope

Xavier

Campêlo

2011

Electronic Notes in Discrete Mathematics

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“…The clique inequality x 4 + x 5 + 4, 5}. Applying Theorem 3.4, we obtain the facet-defining inequality 2x 1 + x 2 + 2x 3 + 2x 4 + x 5 + x 6 + x 7 + x 8 ≤ 3.…”

Section: Theorem 34 Supposementioning

confidence: 91%

“…Since then, other procedures based on graph-theoretical transformations, such as subdividing edges [7], subdividing stars [1], replacing vertices with stars [2] and replacing edges with gears [4], have been described.…”

Section: Introductionmentioning

confidence: 99%

A New Facet Generating Procedure for the Stable Set Polytope

Xavier

Campêlo

2011

Electronic Notes in Discrete Mathematics

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“…For that, it surprisingly turned out that it was not necessary to make use of the description of STAB(G) for claw-free not quasi-line graphs G -with α(G) = 2 (by Cook, see [31]), -with α(G) = 3 (by Pêcher, Wagler [28]), -with α(G) ≥ 4 (by Galluccio, Gentile, Ventura [13,14,15]). …”

Section: Conclusion and Future Researchmentioning

confidence: 99%

Lovász-Schrijver PSD-Operator on Claw-Free Graphs

Bianchi

Escalante

Nasini

et al. 2016

Lecture Notes in Computer Science

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Abstract. The subject of this work is the study of LS+-perfect graphs defined as those graphs G for which the stable set polytope STAB(G) is achieved in one iteration of Lovász-Schrijver PSD-operator LS+, applied to its edge relaxation ESTAB(G). In particular, we look for a polyhedral relaxation of STAB(G) that coincides with LS+(ESTAB(G)) and STAB(G) if and only if G is LS+-perfect. An according conjecture has been recently formulated (LS+-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.

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“…If G is a thickening of

H ∖ X

under a matching

M \subseteq {v_{7} v_{8}}

, then

(G, I (v_{1}) \cup I (v_{2}), I (v_{4}) \cup I (v_{5}))

is a strip, and we say that it is a gear strip , following the terminology of Galluccio et al. . These correspond to thickenings of the class

Z_{5}

in and are a slight generalization of thickenings of XX‐strips as defined in .…”

Section: Compositions Of Strips and Generalized 2‐joinsmentioning

confidence: 99%

Claw‐Free Graphs, Skeletal Graphs, and a Stronger Conjecture on ω, Δ, and χ

King

Reed

2014

Journal of Graph Theory

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The second author's (B.A.R.) ω, Δ, χ conjecture proposes that every graph satisfies χ≤⌈12(Δ+1+ω)⌉. In this article, we prove that the conjecture holds for all claw‐free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way, we discuss a stronger local conjecture, and prove that it holds for claw‐free graphs with a three‐colorable complement. To prove our results, we introduce a very useful χ‐preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so‐called skeletal graphs.

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Gear composition and the stable set polytope

Cited by 16 publications

References 16 publications

A New Facet Generating Procedure for the Stable Set Polytope

A New Facet Generating Procedure for the Stable Set Polytope

Lovász-Schrijver PSD-Operator on Claw-Free Graphs

Claw‐Free Graphs, Skeletal Graphs, and a Stronger Conjecture on ω, Δ, and χ

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