Dominating sets reconfiguration under token sliding

Let $G$ be a graph and $D_s$ and $D_{\textsf{t}}$ be two dominating sets of $G$ of size $k$. Does there exist a sequence $\langle D_0 = D_s, D_1, \ldots, D_{\ell-1}, D_\ell = D_{\textsf{t}} \rangle$ 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 tree-width 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.