The complexity of the Perfect Matching-Cut problem

TL;DR

This paper investigates the computational complexity of the Perfect Matching-Cut problem, establishing NP-completeness for various graph classes and identifying specific classes where polynomial algorithms exist.

Contribution

It provides a comprehensive complexity classification of the Perfect Matching-Cut problem across multiple graph classes, highlighting both hardness and tractability results.

Findings

NP-complete for planar graphs with max degree four

NP-complete for bipartite five-regular graphs

Polynomial algorithms for claw-free, P5-free, diameter two, bipartite diameter three, and bounded tree-width graphs

Abstract

Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three and for bipartite graphs of diameter four. We show that there exist polynomial time algorithms for the following classes of graphs: claw-free, $P_5$-free, diameter two, bipartite with diameter three and graphs with bounded tree-width.