Perfect matching cuts partitioning a graph into complementary subgraphs

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 $\Pi =\mathcal{P}\mathcal{M}$ specifies that the edge cut of the decomposition is a perfect matching. We prove that Comp-Sub($\mathcal{P}\mathcal{M}$) is GI-hard when the graph $G$ is $\{C_{k\ge 7},{\overline{C}}_{k\ge 7}\}$-free. On the other hand, we show that Comp-Sub($\mathcal{P}\mathcal{M}$) is polynomial-time solvable on $hole$-free graphs and on $P_5$-free graphs. Furthermore, we present characterizations of Comp-Sub($\mathcal{P}\mathcal{M}$) on chordal, distance-hereditary, and extended $P_4$-laden graphs.