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Yayın On the maximum cardinality cut problem in proper interval graphs and related graph classes(Elsevier B.V., 2022-01-04) Boyacı, Arman; Ekim, Tınaz; Shalom, MordechaiAlthough 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 the parameterized complexity of this problem. We show that the maximum cardinality cut problem in proper/unit interval graphs is FPT when parameterized by the maximum number of non-empty bubbles in a column of its bubble model. We then generalize this result to a more general graph class by defining new parameters related to the well-known clique-width parameter. Specifically, we define an (?,?,?)-clique-width decomposition of a graph as a clique-width decomposition in which at each step the following invariant is preserved: after discarding at most ? labels, a) every label consists of at most ? sets of twin vertices, and b) all the labels together induce a graph with independence number at most ?. We show that for every two constants ?,?>0 the problem is FPT when parameterized by ? plus the smallest width of an (?,?,?)-clique-width decomposition.Yayın Hierarchical b-Matching(Springer Science and Business Media Deutschland GmbH, 2021) Emek, Yuval; Kutten, Shay; Shalom, Mordechai; Zaks, ShmuelA matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a b-matching every vertex v has an associated bound bv, and a maximum b-matching is a maximum set of edges, such that every vertex v appears in at most bv of them. We study an extension of this problem, termed Hierarchical b-Matching. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. We seek for a maximum set of edges, that obey all bounds (that is, no vertex v participates in more than bv edges, then all the vertices in one subset do not participate in more that subset’s bound of edges, and so on hierarchically). This is a sub-problem of the matroid matching problem which is NP -hard in general. It corresponds to the special case where the matroid is restricted to be laminar and the weights are unity. A pseudo-polynomial algorithm for the weighted laminar matroid matching problem is presented in [8]. We propose a polynomial-time algorithm for Hierarchical b-matching, i.e. the unweighted laminar matroid matching problem, and discuss how our techniques can possibly be generalized to the weighted case.
