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 paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions.

中文翻译：

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关于适当区间图和相关图类中的最大基数切割问题

尽管在两篇不同的论文中声称最大基数切割问题对于适当的区间图是多项式时间可解的，但它们都被证明是错误的。在本文中，当我们引入一些新参数进行参数化时，我们给出了在包含适当区间图和混合单位区间图的图类中的最大基数切割问题的 FPT 算法。这些新参数与适当区间图和混合单位区间图的所谓气泡表示的泛化以及集团宽度分解有关。