On k-club and k-clique numbers in graphs

For a simple undirected graph and a given positive integer k, a k-club is a subset of vertices that induces a subgraph of diameter at most k, and the k-club number ω̄k(G) is the cardinality of a largest k-club in G. In this paper we first prove that for given positive integers k and l, k 6= l, the problem of recognizing whether there is a gap between ω̄k and ω̄l is NP -hard. Then we use this result to show that for k ≥ 2, unless P = NP , one cannot design a polynomial-time algorithm that would detect a k-club of size > ∆(G) + 1 in any graph G with ω̄k(G) > ∆(G) + 1, where ∆(G) denotes the maximum degree of a vertex in G. The same results hold for the maximum k-clique problem as well.