Perfect matching cuts partitioning a graph into complementary subgraphs

TL;DR

This paper investigates the problem of partitioning graphs into two complementary subgraphs with a perfect matching as the edge cut, revealing its computational complexity and providing polynomial-time solutions for specific graph classes.

Contribution

It proves GI-hardness for certain graph classes and offers polynomial algorithms for hole-free and P_5-free graphs, along with characterizations for chordal and related graphs.

Findings

GI-hardness for ree graphs with certain cycles

Polynomial-time solutions for hole-free and P_5-free graphs

Characterizations for chordal, distance-hereditary, and P_4-laden graphs

Abstract

In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $\Pi$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such a way that the edge cut $[V(H),V(\overline{H})]$ satisfies the property $\Pi$. Motivated by previous work, we consider Comp-Sub($\Pi$) when the property $\Pi=\mathcal{PM}$ specifies that the edge cut of the decomposition is a perfect matching. We prove that Comp-Sub($\mathcal{PM}$) is GI-hard when the graph $G$ is $\{C_{k\geq 7}, \overline{C}_{k\geq 7} \}$-free. On the other hand, we show that Comp-Sub($\mathcal{PM}$) is polynomial-time solvable on $hole$-free graphs and on $P_5$-free graphs. Furthermore, we present characterizations of Comp-Sub($\mathcal{PM}$) on chordal, distance-hereditary, and extended $P_4$-laden graphs.