Finding Minimum Matching Cuts in $H$-free Graphs

TL;DR

This paper explores the computational complexity of finding minimum matching cuts in various classes of graphs, providing polynomial algorithms for some classes and NP-hardness results for others, thereby advancing understanding of this problem.

Contribution

It extends polynomial-time algorithms to new graph classes and establishes complexity boundaries, solving open problems and differentiating between minimum and maximum matching cut complexities.

Findings

Polynomial algorithms for $P_8$-free, $S_{1,1,3}$-free, and $(P_6 + P_4)$-free graphs.

NP-hardness of Minimum Matching Cut in $3P_3$-free graphs.

Complexity dichotomies for bounded radius and diameter graphs.

Abstract

A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We investigate the differences in complexity between Minimum Matching Cut, its counterpart Maximum Matching Cut, and the decision problem Matching Cut. Our polynomial-time algorithms for $P_8$-free, $S_{1,1,3}$-free and $(P_6 + P_4)$-free graphs extend the cases where Minimum Matching Cut and Maximum Matching Cut are known to differ in complexity. In addition, they solve open cases for the well-studied problem Matching Cut. The NP-hardness proof for $3P_3$-free graphs implies that Minimum Matching Cut and Matching Cut, which is polynomial-time solvable even for $sP_3$-free graphs, for any $s \geq 1$, differ in complexity on certain graph classes. Further, we give complexity dichotomies for both general and bipartite graphs of bounded radius and diameter.