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Communication Dans Un Congrès Année : 2023

Matching Cuts in Graphs of High Girth and H-Free Graphs

Daniël Paulusma

Résumé

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed d ≥ 1, is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs.
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Dates et versions

hal-04632864 , version 1 (02-07-2024)

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Carl Feghali, Felicia Lucke, Daniël Paulusma, Bernard Ries. Matching Cuts in Graphs of High Girth and H-Free Graphs. ISAAC 2023, Dec 2023, Kyoto (Japan), Japan. ⟨10.4230/LIPIcs.ISAAC.2023.31⟩. ⟨hal-04632864⟩
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