Computing Subset Transversals in $H$-Free Graphs

TL;DR

This paper investigates the computational complexity of Subset Feedback Vertex Set and Subset Odd Cycle Transversal problems within H-free graphs, providing a comprehensive classification except for a specific graph case.

Contribution

It determines the complexity of these problems on H-free graphs for all H except one, and introduces the polynomial-time solvability of Subset Vertex Cover for certain H-free graphs.

Findings

Complexity classification for Subset Feedback Vertex Set and Subset Odd Cycle Transversal on H-free graphs.

Polynomial-time algorithm for Subset Vertex Cover on (sP_1+P_4)-free graphs.

Complete complexity results except for the case H=sP_1+P_4 with s≥1.

Abstract

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $s\geq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $s\geq 1$.