Computing Weighted Subset Transversals in $H$-Free Graphs

TL;DR

This paper establishes a near-complete complexity classification for the Weighted Subset Odd Cycle Transversal problem in H-free graphs, extending to related problems like Feedback Vertex Set, with implications for graph algorithms.

Contribution

It provides a comprehensive complexity dichotomy for Weighted Subset Odd Cycle Transversal in H-free graphs and generalizes recent results to related problems.

Findings

Established a near-complete complexity dichotomy for the problem.

Extended the approach to Weighted Subset Feedback Vertex Set.

Generalized recent results in the area.

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. Our general approach can also be used for Weighted Subset Feedback Vertex Set, which enables us to generalize a recent result of Papadopoulos and Tzimas.