Abstract

A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time for \((sP_1+P_3)\)-free graphs for every integer \(s\ge 1\). We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. For the latter two problems we also prove that they are polynomial-time solvable for cographs; this was known already for Feedback Vertex Set and Odd Cycle Transversal.

Highlights

  • In this paper we focus on proving new complexity results for Feedback Vertex Set, Connected Feedback Vertex Set, Odd Cycle Transversal and Connected Odd Cycle Transversal for H-free graphs

  • We will prove that Feedback Vertex Set and Odd Cycle Transversal and their connected variants can be solved in polynomial time for-free graphs

  • We proved polynomial results for Feedback Vertex Set and Odd Cycle Transversal and their connected variants for H-free graphs, where H = P4 or H = sP1 + P3

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Summary

Introduction

When H contains a cycle or a claw, the problems Connected Vertex Cover [25], Feedback Vertex Set (respectively, via a folklore trick, see [3,23], and due to hardness for the subclass of line graphs of planar cubic bipartite graphs [25]), Connected Feedback Vertex Set [8], Odd Cycle Transversal [8] and Connected Odd Cycle Transversal [8] are all NPcomplete for H-free graphs For these five problems, we are left to consider only the case where H is a linear forest. The Feedback Vertex Set and Odd Cycle Transversal problems are polynomial-time solvable for permutation graphs [4], and for P4-free graphs. For every s ≥ 1, both problems and their connected variants are polynomial-time solvable for sP2-free graphs [8], using the price of connectivity for feedback vertex set [2,19].4

Our Results
Preliminaries
Conclusions

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