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Matching cut Perfect matching -free graph Diameter Radius DichotomyThe (Perfect) Matching Cut problem is to decide if a graph has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of .Both Matching Cut and Perfect Matching Cut are known to be NP-complete.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 perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph.Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and -free graphs.A disconnected perfect matching of a graph is a perfect matching that contains a matching cut of .We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes.In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for ( 6 + 2 )-free graphs for every ≥ 0, extending a known result for 5 -free graphs (Bouquet and Picouleau, 2020).
Lucke et al. (Thu,) studied this question.