# Computing Subset Vertex Covers in H-Free Graphs

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 ⊆ 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 ≠ NP). We also prove new polynomial time results. We first give a dichotomy on graphs where G[T] is H-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs G, for which G[T] is H-free, if H = sP_1 + tP_2 and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for (sP_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.

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