Finding Matching Cuts in $H$-Free Graphs

TL;DR

This paper investigates the computational complexity of the Matching Cut problem and its variants in $H$-free graphs, establishing new NP-completeness results and providing comprehensive complexity classifications for these problems.

Contribution

It presents new NP-completeness results for Matching Cut variants in $H$-free graphs, including addressing an open question about $P_r$-free graphs.

Findings

NP-completeness of Matching Cut in $P_r$-free graphs for some $r>0$

Complexity classifications for Matching Cut variants in $H$-free graphs

Resolution of an open problem from Bouquet and Picouleau (2020)

Abstract

The NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on $H$-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant $r>0$ such that the first variant is NP-complete for $P_r$-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for $H$-free graphs.