Gear Composition of Stable Set Polytopes and G-Perfection - Galluccio, Anna; Gentile, Claudio; Ventura, Paolo (Mathematics of operations research, 2009)
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( H e ), where H e is the graph obtained from H by subdividing the edge e . We also introduce the class of -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 -perfect, provided that any graph obtained from H by subdividing a subset of its simplicial edges is -perfect. In particular, we show that a large subclass of claw-free graphs is -perfect.