The Parameterised Complexity of List Problems on Graphs of Bounded Treewidth

TL;DR

This paper investigates the parameterised complexity of various list problems on graphs with bounded treewidth, establishing fixed parameter tractability for some problems and W[1]-hardness for others, thus resolving open questions.

Contribution

It provides a comprehensive complexity classification of list problems on graphs of bounded treewidth and pathwidth, resolving two open questions from prior research.

Findings

List Edge Chromatic Number is fixed parameter tractable by treewidth.

List Total Chromatic Number is fixed parameter tractable by treewidth.

List Hamilton Path is W[1]-hard even when parameterised by pathwidth.

Abstract

We consider the parameterised complexity of several list problems on graphs, with parameter treewidth or pathwidth. In particular, we show that List Edge Chromatic Number and List Total Chromatic Number are fixed parameter tractable, parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even parameterised by pathwidth. These results resolve two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).