Finding clubs in graph classes

Abstract

For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. We study a relation of clubs and cliques. For a positive integer s, we say that a graph class G has the s-clique-power property if for every graph G∈G, every maximal clique in Gs is an s-club in G. Our main combinatorial results show that both 4-chordal graphs and AT-free graphs have the s-clique-power property for all s≥2. This has various algorithmic consequences. In particular we show that a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s≥2, which gives an interesting contrast to the fact that the problem is NP-hard for s=1 on this graph class.