Classifying subset feedback vertex set for H-free graphs

Giacomo Paesani, Daniël Paulusma, Pawel Rzazewski

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<math><mi is="true">w</mi><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo stretchy="false" is="true">)</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math> 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.