On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

TL;DR

This paper corrects previous misconceptions by providing fixed-parameter tractable algorithms for the maximum cardinality cut problem in proper interval and related graphs, using novel parameters linked to graph representations.

Contribution

It introduces new parameters based on bubble representations and clique-width decompositions, enabling FPT algorithms for the problem in broader graph classes.

Findings

FPT algorithms for proper interval graphs

Extension to mixed unit interval graphs

Parameters related to bubble representations and clique-width

Abstract

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.