Computing subset vertex covers in H-free graphs

Abstract

We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph $G=(V,E)$, a subset $T\subseteq V$ and integer k, if V has a subset S of size at most k, such that S contains at least one end-vertex of every edge incident to a vertex of T. A graph is H-free if it does not contain H as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw, diamond)-free planar graphs and on 2-unipolar graphs, a subclass of $2P_3$-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under $P\ne \mathrm{NP}$). We also prove new polynomial time results, some of which follow from a reduction to Vertex Cover restricted to classes of probe graphs. We first give a dichotomy on graphs where $G\left[T\right]$ is H-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs G, for which $G\left[T\right]$ is H-free, if $H=s{P}_1+t{P}_2$ and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for $(s{P}_1+P_2+P_3)$-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on H-free graphs.