Computing Weighted Subset Odd Cycle Transversals in H-free graphs

Abstract

For the Odd Cycle Transversal problem, the task is to find a small set S of vertices in a graph that intersects every cycle of odd length. The Subset Odd Cycle Transversal problem requires S to intersect only those odd cycles that include a vertex of a distinguished vertex subset T. If we are given weights for the vertices, we ask instead that S has small weight: this is the problem Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity dichotomy for Weighted Subset Odd Cycle Transversal for graphs that do not contain a graph H as an induced subgraph. In particular, our result shows that the complexities of the weighted and unweighted variant do not align on H-free graphs, just as Papadopoulos and Tzimas showed for Subset Feedback Vertex Set.