Clubs are generalizations of cliques. 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. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, 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 a superclass of these graphs, 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 on this class 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, a superclass of weakly 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. © 2013 Springer-Verlag.
CITATION STYLE
Golovach, P. A., Heggernes, P., Kratsch, D., & Rafiey, A. (2013). Cliques and clubs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7878 LNCS, pp. 276–287). https://doi.org/10.1007/978-3-642-38233-8_23
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