Abstract
In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph \(G=(V,E)\), and an edge set property \(\varPi \), 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 \(\varPi \). Motivated by previous work, we consider \(\textsc {Comp}\)-\(\textsc {Sub}(\varPi )\) when the property \(\varPi =\mathcal {PM}\) specifies that the edge cut of the decomposition is a perfect matching. We prove that \(\textsc {Comp}\)-\(\textsc {Sub}(\mathcal {PM})\) is \({\mathsf {GI}}\)-hard when the graph G is \(\{C_{k\ge 7}, \overline{C}_{k\ge 7} \}\)-free. On the other hand, we show that \(\textsc {Comp}\)-\(\textsc {Sub}(\mathcal {PM})\) is polynomial time solvable on hole-free graphs and on \(P_5\)-free graphs. Furthermore, we present characterizations of \(\textsc {Comp}\)-\(\textsc {Sub}(\mathcal {PM})\) on chordal, distance-hereditary, and extended \(P_4\)-laden graphs.
This research has received funding from Rio de Janeiro Research Support Foundation (FAPERJ) under grant agreement E-26/201.344/2021, National Council for Scientific and Technological Development (CNPq) under grant agreement 309832/2020-9, and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement CUTACOMBS (No. 714704).
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Castonguay, D., Coelho, E.M.M., Coelho, H., Nascimento, J., Souza, U.S. (2022). Perfect Matching Cuts Partitioning a Graph into Complementary Subgraphs. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_19
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