The Parameterized Complexity of s -Club with Triangle and Seed Constraints

The s - Club problem asks whether a given undirected graph 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.