Computer Science > Data Structures and Algorithms
[Submitted on 14 Jan 2022 (v1), last revised 20 Jun 2022 (this version, v2)]
Title:The Parameterized Complexity of s-Club with Triangle and Seed Constraints
View PDFAbstract: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 $\ell$ 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.
Submission history
From: Frank Sommer [view email][v1] Fri, 14 Jan 2022 20:10:37 UTC (64 KB)
[v2] Mon, 20 Jun 2022 11:32:45 UTC (49 KB)
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