The Parameterized Complexity of s-Club with Triangle and Seed Constraints
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The s-Club problem asks, for a given undirected graph G, whether G contains a vertex set S of size at least k such that G[S], the subgraph of G induced by S, has diameter at most s. We consider variants of s-Club where one additionally demands that each vertex of G[S] is contained in at least ℓ triangles in G[S], that each edge of G[S] is contained in at least ℓ triangles in G[S], or that S contains a given set W of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size k, making them significantly harder than the unconstrained s-Club problem. On the positive side, we obtain some FPT algorithms for the case when ℓ=1 and for the case when G[W], the graph induced by the set of seed vertices, is a clique.
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