Perfect matching cuts partitioning a graph into complementary subgraphs
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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 H, for some graph H, in such a way that the edge cut [V(H),V(H)] 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 {C_k≥ 7, C_k≥ 7}-free. On the other hand, we show that Comp-Sub(𝒫ℳ) is polynomial-time solvable on hole-free graphs and on P_5-free graphs. Furthermore, we present characterizations of Comp-Sub(𝒫ℳ) on chordal, distance-hereditary, and extended P_4-laden graphs.
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