The Complexity of Distance-$r$ Dominating Set Reconfiguration

For a fixed integer $r \geq 1$, a distance-$r$ dominating set of a graph $G = (V, E)$ is a vertex subset $D \subseteq V$ such that every vertex in $V$ is within distance $r$ from some member of $D$. Given two distance-$r$ dominating sets $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of distance-$r$ dominating sets that transforms $D_s$ into $D_t$ (or vice versa) such that each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. The problem for $r = 1$ has been well-studied in the literature. We consider D$r$DSR for $r \geq 2$ under two well-known reconfiguration rules: Token Jumping ($\mathsf{TJ}$, which involves replacing a member of the current D$r$DS by a non-member) and Token Sliding ($\mathsf{TS}$, which involves replacing a member of the current D$r$DS by an adjacent non-member). We show that D$r$DSR ($r \geq 2$) is $\mathtt{PSPACE}$-complete under both $\mathsf{TJ}$ and $\mathsf{TS}$ on bipartite graphs, planar graphs of maximum degree six and bounded bandwidth, and chordal graphs. On the positive side, we show that D$r$DSR ($r \geq 2$) can be solved in polynomial time on split graphs and cographs under both $\mathsf{TS}$ and $\mathsf{TJ}$ and on trees and interval graphs under $\mathsf{TJ}$. Along the way, we observe some properties of a shortest reconfiguration sequence in split graphs when $r = 2$, which may be of independent interest.