We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of CLIQUE: $s$-CLUB and $s$-CLIQUE, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and $\gamma$-COMPLETE SUBGRAPH in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that $s$-CLUB and $s$-CLIQUE are NP-hard even restricted to graphs of degeneracy $\le 3$ whenever $s \ge 3$, and to graphs of degeneracy $\le 2$ whenever $s \ge 5$, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. We also obtain that these problems are solvable in polynomial time on graphs of degeneracy $1$. Concerning $\gamma$-COMPLETE SUBGRAPH, we prove that it is W[1]-hard parameterized by both the degeneracy, which implies the W[1]-hardness parameterized by the number of vertices in the $\gamma$-complete-subgraph, and the number of elements outside the $\gamma$-complete subgraph.

翻译：我们研究了图聚类识别若干问题的参数复杂度。换言之，我们探讨图是否包含足够大且高度连通的子图。本文研究了团问题的三种松弛形式：$s$-CLUB 和 $s$-CLIQUE（松弛聚焦于分别对应子图内距离和原图内距离），以及 $\gamma$-COMPLETE SUBGRAPH（松弛聚焦于子图中最小度数）。由于这三个问题已知为 NP-难问题，我们在此研究它们的参数复杂度。我们证明：当 $s \ge 3$ 时，即使在退化度 $\le 3$ 的图限制下，$s$-CLUB 和 $s$-CLIQUE 仍是 NP-难的；当 $s \ge 5$ 时，即使在退化度 $\le 2$ 的图限制下也是如此——该结果严格强于其以退化度为参数的 W[1]-难性。我们还发现这些问题在退化度为 $1$ 的图上可在多项式时间内求解。对于 $\gamma$-COMPLETE SUBGRAPH，我们证明其以退化度为参数是 W[1]-难的，这蕴含了以 $\gamma$-完全子图顶点数及位于该子图外部的元素数量为参数的 W[1]-难性。