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 \(\ell \) triangles in G[S], that each edge of G[S] is contained in at least \(\ell \) 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 \(\ell =1\) and for the case when G[W], the graph induced by the set of seed vertices, is a clique.

s - Club问题询问给定的无向图 G是否包含大小至少为k的顶点集 S ，使得 G [ S ] （由 S导出的G的子图）的 直径至多为 s。我们考虑s - Club的变体，其中还要求 G [ S ] 的每个顶点至少包含在G [ S ] 中的\(\ell \) 个三角形 中，并且 G [ S ]的每条边] 至少包含在G [ S ] 中的\(\ell \) 个三角形 中，或者 S包含给定的 种子顶点集W。我们表明，一般来说，当通过解大小 k参数化时，这些变体是 W[1]-困难的，这使得它们比无约束的s - Club问题要困难得多 。从积极的一面来看，我们获得了一些 FPT 算法，用于当 \(\ell =1\)时的情况和当 G [ W ]（由种子顶点集导出的图）是团时的情况。