The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r > 0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.
@InProceedings{lucke_et_al:LIPIcs.ISAAC.2022.22, author = {Lucke, Felicia and Paulusma, Dani\"{e}l and Ries, Bernard}, title = {{Finding Matching Cuts in H-Free Graphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.22}, URN = {urn:nbn:de:0030-drops-173076}, doi = {10.4230/LIPIcs.ISAAC.2022.22}, annote = {Keywords: matching cut, perfect matching, H-free graph, computational complexity} }
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