On the maximum cardinality cut problem in proper interval graphs and related graph classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this work we consider parameterized complexity problem. We show proper/unit graphs FPT when by number non-empty bubbles a column its bubble model. then generalize result more general graph class defining new parameters related well-known clique-width parameter. Specifically, define an (α,β,δ)-clique-width decomposition as which at each step following invariant preserved: after discarding most δ labels, a) every label consists β sets twin vertices, and b) all labels together induce with independence α. constants α,δ>0 plus smallest width decomposition.