Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Abstract

In the Feedback Vertex Set problem, we aim to find a small set S of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires S to intersect only those cycles that include a vertex of some specified set T . We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex u has weight w(u) > 0 and we ask that S has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for H-free graphs, that is, graphs that do not contain a graph H as an induced subgraph.