Feedback Vertex Set on Hamiltonian Graphs

TL;DR

This paper investigates the complexity of the Feedback Vertex Set problem within specific subclasses of Hamiltonian graphs, demonstrating NP-hardness even under restrictive conditions such as regularity, planarity, and known Hamiltonian cycles.

Contribution

It establishes NP-hardness of Feedback Vertex Set on various restricted classes of Hamiltonian graphs, including p-Hamiltonian-ordered graphs, expanding understanding of problem complexity.

Findings

Feedback Vertex Set is NP-hard on regular Hamiltonian graphs.

Feedback Vertex Set remains NP-hard on planar and regular Hamiltonian graphs.

NP-hardness persists even when a Hamiltonian cycle is provided as input.

Abstract

We study the computational complexity of Feedback Vertex Set on subclasses of Hamiltonian graphs. In particular, we consider Hamiltonian graphs that are regular or are planar and regular. Moreover, we study the less known class of $p$-Hamiltonian-ordered graphs, which are graphs that admit for any $p$-tuple of vertices a Hamiltonian cycle visiting them in the order given by the tuple. We prove that Feedback Vertex Set remains NP-hard in these restricted cases, even if a Hamiltonian cycle is additionally given as part of the input.