A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter

TL;DR

This paper establishes the computational complexity of the Matching Cut problem in graphs of fixed diameter, proving NP-completeness for diameters three and above, while providing polynomial algorithms for diameters two and three.

Contribution

It resolves an open problem by proving NP-completeness of Matching Cut for diameter d ≥ 3 and bipartite graphs of diameter d ≥ 4, and offers efficient algorithms for diameters 2 and 3.

Findings

Matching Cut is NP-complete for graphs of diameter d ≥ 3.

Matching Cut is NP-complete for bipartite graphs of diameter d ≥ 4.

Polynomial-time algorithms exist for graphs of diameter 2 and 3.

Abstract

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-J\'ozefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (2008) 574-582]. We then show that, for any fixed integer $d\ge 4$, Matching Cut is NP-complete even when restricted to the class of bipartite graphs of diameter $d$. Complementing the hardness results, we show that Matching Cut is polynomial-time solvable in the class of bipartite graphs of diameter at most three, and point out a new and simple polynomial-time algorithm solving Matching Cut in graphs of diameter $2$.