Reconfiguration of Cliques in a Graph

TL;DR

This paper investigates the reconfiguration of cliques in graphs, establishing complexity results and polynomial algorithms for specific graph classes, thereby advancing understanding of clique transformations.

Contribution

It proves the equivalence of three reconfiguration rules for cliques and provides complexity and polynomial-time results for various graph classes.

Findings

Reconfiguration rules are equivalent for cliques.

Reconfiguration problems are PSPACE-complete for perfect graphs.

Polynomial algorithms exist for certain graph classes like chordal and bipartite graphs.

Abstract

We study reconfiguration problems for cliques in a graph, which determine whether there exists a sequence of cliques that transforms a given clique into another one in a step-by-step fashion. As one step of a transformation, we consider three different types of rules, which are defined and studied in reconfiguration problems for independent sets. We first prove that all the three rules are equivalent in cliques. We then show that the problems are PSPACE-complete for perfect graphs, while we give polynomial-time algorithms for several classes of graphs, such as even-hole-free graphs and cographs. In particular, the shortest variant, which computes the shortest length of a desired sequence, can be solved in polynomial time for chordal graphs, bipartite graphs, planar graphs, and bounded treewidth graphs.