Dominating sets reconfiguration under token sliding

TL;DR

This paper studies the reconfiguration problem of dominating sets in graphs, proving PSPACE-completeness in general but polynomial-time solvability in certain graph classes like dually chordal graphs and cographs.

Contribution

It establishes the computational complexity of dominating set reconfiguration under token sliding, showing hardness in some graph classes and efficient algorithms in others.

Findings

PSPACE-complete for split, bipartite, and bounded treewidth graphs

Polynomial-time algorithm for dually chordal graphs and cographs

Reconfiguration problem complexity varies significantly across graph classes

Abstract

Let $G$ be a graph and $D_s$ and $D_t$ be two dominating sets of $G$ of size $k$. Does there exist a sequence $\langle D_0 = D_s, D_1, \ldots, D_{\ell-1}, D_\ell = D_t \rangle$ of dominating sets of $G$ such that $D_{i+1}$ can be obtained from $D_i$ by replacing one vertex with one of its neighbors? In this paper, we investigate the complexity of this decision problem. We first prove that this problem is PSPACE-complete, even when restricted to split, bipartite or bounded treewidth graphs. On the other hand, we prove that it can be solved in polynomial time on dually chordal graphs (a superclass of both trees and interval graphs) or cographs.