On (in)tractability of connection and cut problems

Abstract

This work addresses connection and cut problems from the viewpoint of graph classes and computational complexity, classic and parameterized. Regarding connection problems, we investigate the so-called TERMINAL CONNECTION problem (TCP), which can be seen as a generalisation of the classical STEINER TREE problem. We propose several complexity results for TCP, when restricted to specific graph classes, and some of its input parameters are fixed. As for cut problems, we analyse the complexity of the classical MAXCUT problem. We introduce the first complexity classification for the problem on interval graphs of bounded interval count. In addition, we prove the NP-completeness of MAXCUT on permutation graphs, settling a question posed by David S. Johnson in the Ongoing Guide to NP-completeness, which has been open since 1985. Finally, we consider the problem of computing the zig-zag number of a directed graph, which is a directed width measure defined over cuts of a graph.