Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius
The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P₆+sP₂)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P₃-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs.
matching cut
perfect matching
H-free graph
diameter
radius
dichotomy
Mathematics of computing~Graph algorithms
64:1-64:15
Regular Paper
Felicia
Lucke
Felicia Lucke
Department of Informatics, University of Fribourg, Switzerland
https://orcid.org/0000-0002-9860-2928
Daniël
Paulusma
Daniël Paulusma
Department of Computer Science, Durham University, UK
https://orcid.org/0000-0001-5945-9287
Bernard
Ries
Bernard Ries
Department of Informatics, University of Fribourg, Switzerland
https://orcid.org/0000-0003-4395-5547
10.4230/LIPIcs.MFCS.2023.64
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Felicia Lucke, Daniël Paulusma, and Bernard Ries
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