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

TL;DR

This paper establishes the computational complexity boundaries for the Subset Feedback Vertex Set problem and its weighted variant in $H$-free graphs, combining known NP-hardness with new polynomial-time results.

Contribution

It provides a complete complexity classification (dichotomy) for these problems on $H$-free graphs, advancing understanding of their computational tractability.

Findings

Full complexity dichotomies for Subset Feedback Vertex Set and weighted version.

Identification of polynomial-time solvable cases in $H$-free graphs.

Confirmation of NP-hardness for other cases.

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.