The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs

TL;DR

This paper investigates the complexity of reconfiguring list colorings in graphs, showing PSPACE-completeness for certain classes and polynomial-time solutions for graphs with low pathwidth, thus clarifying the problem's complexity landscape.

Contribution

It establishes the PSPACE-completeness of the list coloring reconfiguration problem for bipartite series-parallel graphs and provides a polynomial-time algorithm for graphs with pathwidth one.

Findings

PSPACE-complete for bipartite series-parallel graphs

Polynomial-time algorithm for graphs with pathwidth one

Complexity varies with graph pathwidth

Abstract

We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACE-completeness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives precise analyses of the problem with respect to pathwidth.