Reconfiguring Independent Sets in Cographs

TL;DR

This paper presents efficient algorithms to analyze the connectivity of reconfiguration graphs of independent sets in cographs, solving an open problem and enabling quick connectivity checks.

Contribution

It provides the first cubic-time algorithm for connectivity in TAR_k(G) for cographs and a linear-time method for component membership testing.

Findings

Decides connectivity of TAR_k(G) in cubic time for cographs.

Provides a linear-time algorithm to determine if two independent sets are in the same component.

Solves an open problem in graph reconfiguration theory.

Abstract

Two independent sets of a graph are adjacent if they differ on exactly one vertex (i.e. we can transform one into the other by adding or deleting a vertex). Let $k$ be an integer. We consider the reconfiguration graph $TAR_k(G)$ on the set of independent sets of size at least $k$ in a graph $G$, with the above notion of adjacency. Here we provide a cubic-time algorithm to decide whether $TAR_k(G)$ is connected when $G$ is a cograph, thus solving an open question of~[Bonsma 2014]. As a by-product, we also describe a linear-time algorithm which decides whether two elements of $TAR_k(G)$ are in the same connected component.