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 Π, and asked whether G can be decomposed into two graphs, H and its complement H̄, for some graph H, in such a way that the edge cut [V(H), V(H̄)] satisfies the property Π. Motivated by previous work, we consider COMP-SUB(Π) when the property Π=PM specifies that the edge cut of the decomposition is a perfect matching. We prove that COMP-SUB(PM) is GI-hard when the graph G is C5-free or G is {Ck ≥ 7, C̄k ≥ 7}-free. On the other hand, we show that COMP-SUB(PM) is polynomial-time solvable on hole-free graphs and on P5-free graphs. Furthermore, we present characterizations of COMP-SUB(PM) on chordal, distance-hereditary, and extended P4-laden graphs.