2-clique-bond of stable set polyhedra

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.We consider a graph G obtained as the 2-clique-bond of two graphs G 1 and G 2 and we study the polyhedral properties of the stable set polytope associated with this graph. In particular, 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 four graphs that are related to G 1 and G 2 .We show how to apply the 2-clique-bond composition to provide the complete linear description of large classes of graphs.

Stable sets and graphs with no even holes

“…Since Z is a clique and G is not a near-clique, we have that load(G, Z) ≥ 2. So, by (12), at least one of (G 1 , Z) and (G 2 , U) has positive load. So a normal decomposition in G gives a list H with:…”

“…For those G 1 has all nodes in Z, so U is a clique. Hence (12) gives: load(G 2 , U) = load(G, Z), which is positive.…”

“…Faenza, Oriolo, and Stauffer [11] used strip-structures to obtain extended formulations and polynomial-time algorithms for stable sets problems in "claw-free" graphs. The "2-cliquebonds" that Galluccio, Gentile, and Ventura [12] use to compose linear formulations of stable set problems are generalized amalgam separations as well.…”