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arXiv · 2201.05654

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

Jaroslav Garvardt·Christian Komusiewicz·Frank Sommer
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Abstract

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.

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Jaroslav Garvardt, Christian Komusiewicz, Frank Sommer. 2022-01-14. The Parameterized Complexity of s-Club with Triangle and Seed Constraints. https://arxiv.org/abs/2201.05654

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