Reconfiguration of Colorable Sets in Classes of Perfect Graphs

TL;DR

This paper investigates the complexity of reconfiguring c-colorable vertex sets in perfect graphs, providing efficient algorithms for some classes and proving hardness results for others, advancing understanding of graph reconfiguration problems.

Contribution

It offers a combinatorial characterization and linear-time algorithm for reconfiguration in interval graphs, and establishes PSPACE-completeness results for chordal and co-comparability graphs.

Findings

Linear-time algorithm for shortest reconfiguration sequences in interval graphs

PSPACE-completeness of reachability in chordal and co-comparability graphs

Polynomial-time solvability for split graphs with fixed c

Abstract

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.