Edge clique cover of claw-free graphs

Clique or hole in claw-free graphs

Given a claw-free graph and two non-adjacent vertices x and y without common neighbours we prove that there exists a hole through x and y unless the graph contains the obvious obstruction, namely a clique separating x and y. We derive two applications: We give a necessary and sufficient condition for the existence of an induced x–z path through y, where x, y, z are prescribed vertices in a claw...

Clique minors in claw-free graphs

Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα ≥ n, Hadwiger’s conjecture implies that G has a clique minor of size n α . In this paper we prove that this is true for connected claw-free graphs with α ≥ 3. We also show that this result is tight by pro...

Minimum Clique Cover in Claw-Free Perfect Graphs and the Weak Edmonds-Johnson Property

We give new algorithms for the minimum (weighted) clique cover in a claw-free perfect graph G, improving the complexity from O(|V (G)|) to O(|V (G)|). The new algorithms build upon neat reformulations of the problem: it basically reduces either to solving a 2-SAT instance (in the unweighted case) or to testing if a polyhedra associated with the edge-vertex incidence matrix of a bidirected graph...

Triangulation and Clique Separator Decomposition of Claw-Free Graphs

Finding minimal triangulations of graphs is a well-studied problem with many applications, for instance as first step for efficiently computing graph decompositions in terms of clique separators. Computing a minimal triangulation can be done in O(nm) time and much effort has been invested to improve this time bound for general and special graphs. We propose a recursive algorithm which works for...

Generalized clique family inequalities for claw-free graphs