On The Complexity of Matching Cut for Graphs of Bounded Radius and $H$-Free Graphs

TL;DR

This paper establishes a complexity dichotomy for the Matching Cut problem in graphs of bounded radius and $H$-free graphs, showing polynomial solvability for certain classes and NP-completeness for others, extending previous results.

Contribution

It proves a dichotomy for Matching Cut complexity based on graph radius and $H$-free conditions, extending polynomial-time solvability results to broader graph classes.

Findings

Polynomial-time solvability for $P_6$-free graphs

Extension to $(sP_3+P_6)$-free graphs for all $s",

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Abstract

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free graphs. We then extend our result to hold even for $(sP_3+P_6)$-free graphs for every $s\geq 0$ and initiate a complexity classification of Matching Cut for $H$-free graphs.