Gear Composition of Stable Set Polytopes and G-Perfection

Authors:
Anna Galluccio;Claudio Gentile;Paolo Ventura
Affiliations:
Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti,” CNR, 00185 Rome, Italy;Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti,” CNR, 00185 Rome, Italy;Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti,” CNR, 00185 Rome, Italy
Venue:
Mathematics of Operations Research
Year:
2009

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Abstract

Graphs obtained by applying the gear composition to a given graph H are called geared graphs. We show how a linear description of the stable set polytope STAB(G) of a geared graph G can be obtained by extending the linear inequalities defining STAB(H) and STAB(He), where He is the graph obtained from H by subdividing the edge e. We also introduce the class of G-perfect graphs, i.e., graphs whose stable set polytope is described by nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, and some special inequalities called geared inequalities and g-lifted inequalities. We prove that graphs obtained by repeated applications of the gear composition to a given graph H are G-perfect, provided that any graph obtained from H by subdividing a subset of its simplicial edges is G-perfect. In particular, we show that a large subclass of claw-free graphs is G-perfect.