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

TL;DR

This paper investigates the computational complexity of constrained s-Club problems in graphs, revealing W[1]-hardness in general but providing fixed-parameter tractable algorithms for specific cases.

Contribution

It introduces new variants of the s-Club problem with triangle and seed constraints and analyzes their parameterized complexity, establishing hardness results and FPT algorithms.

Findings

Variants are W[1]-hard when parameterized by solution size.

FPT algorithms exist when $oldsymbol{ extit{ ext{l}}=1}$.

FPT algorithms also exist when seed vertices induce a clique.

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.