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中几个问题的参数化复杂性。换言之，我们探究图是否包含一个足够大且充分连通的子图。本文研究了团簇的三种松弛形式：$s$-俱乐部和$s$-团簇（其松弛聚焦于簇内部和原始图中的距离），以及$\gamma$-完全子图（其松弛聚焦于簇内部的最小度）。由于这三个问题已知为NP困难，我们在此研究其参数化复杂性。我们证明：当$s \ge 3$时，$s$-俱乐部和$s$-团簇即使在退化度$\le 3$的图中也是NP困难的；当$s \ge 5$时，它们在退化度$\le 2$的图中同样NP困难——这一结果严格强于以退化度为参数的W[1]-困难性。此外，我们得到这些问题在退化度为$1$的图上可在多项式时间内求解。关于$\gamma$-完全子图，我们证明其以退化度为参数是W[1]-困难的（这表明以$\gamma$-完全子图的顶点数为参数也是W[1]-困难），同时以$\gamma$-完全子图外部元素数量为参数也是W[1]-困难的。