The Perfect Matching Cut Problem Revisited

TL;DR

This paper investigates the computational complexity of the Perfect Matching Cut problem, proving its NP-completeness in certain graph classes, identifying polynomial cases, and providing exponential time algorithms under the Exponential Time Hypothesis.

Contribution

It establishes NP-completeness of PMC in bipartite graphs with degree 3 and large girth, and introduces polynomial algorithms for specific graph classes, along with exponential algorithms under ETH.

Findings

PMC is NP-complete for bipartite graphs with max degree 3 and large girth.

Polynomial-time solutions exist for claw-free graphs and graphs without a 5-vertex path.

An exact algorithm solves PMC in O*(1.2721^n) time.

Abstract

In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that PMC remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which PMC is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no $O^*(2^{o(n)})$-time algorithm for PMC even when restricted to $n$-vertex bipartite graphs, and also show that PMC can be solved in $O^*(1.2721^n)$ time by means of an exact branching algorithm.