Computational complexity of classical problems for hereditary clique-helly graphs

Abstract

A graph is clique-Helly when its cliques satisfy the Helly property. A graph is hereditary clique-Helly when every induced subgraph of it is clique-Helly. The decision problems associated to the stability, chromatic, clique and clique-covering numbers are NP-complete for clique-Helly graphs. In this note, we analyze the complexity of these problems for hereditary clique-Helly graphs. Some of them can be deduced easily by known results. We prove that the clique-covering problem remains NP-complete for hereditary clique-Helly graphs. Furthermore, the decision problems associated to the clique-transversal and the clique-independence numbers are analyzed too. We prove that they remain NP-complete for a smaller class: hereditary clique-Helly split graphs.

On hereditary clique-Helly self-clique graphs

“…In terms of Szwarcfiter's characterization, HCH graphs are those for which not only every extended triangle has a universal vertex, but such a vertex exists even in the original triangle. See [5] for a recent work on HCH graphs.…”