Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters

TL;DR

This paper investigates the fixed-parameter tractability of the list coloring reconfiguration problem, providing algorithms for certain parameters and proving hardness results for others, advancing understanding of its computational complexity.

Contribution

The paper introduces fixed-parameter algorithms for the list coloring reconfiguration problem based on graph parameters and establishes W[1]-hardness results for specific cases.

Findings

Fixed-parameter algorithm for k and modular-width

Fixed-parameter algorithm for shortest variant with vertex cover size

W[1]-hardness when parameterized only by vertex cover size

Abstract

Let $G$ be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of $k$ colors. For two given list colorings of $G$, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant $k$. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by $k$ and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant when parameterized by $k$ and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only $k$ is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.