The Complexity of Distance-$r$ Dominating Set Reconfiguration

TL;DR

We study Distance-$r$ Dominating Set Reconfiguration (D\$r$DSR) for fixed \$r \ge 2\$ under Token Sliding (\$\mathsf{TS}\$) and Token Jumping (\$\mathsf{TJ}\$). The paper establishes a complexity dichotomy: polynomial-time solvability on split graphs (and, more broadly, on trees under certain rules and on cographs/dually chordal graphs) versus \$\mathsf{PSPACE}\$-hardness on planar graphs with maximum degree three and bounded bandwidth, with extended hardness for chordal and bipartite graph classes when \$r \ge 2\$. It provides tight bounds on the length of shortest reconfiguration sequences on split graphs for \$r=2\$ and delivers a linear-time TJ algorithm on trees, along with a linear-time construction that reduces D\$r$DSR to canonical forms to enable fast reconfigurations. The results employ graph-power reductions, canonical D\$r$DS constructions, and NCL-based reductions to map between standard reconfiguration problems and D\$r$DSR, clarifying the boundary between tractable and intractable instances and leaving open questions for \$\mathsf{TS}\$ on trees and interval graphs.

Abstract

For a fixed integer $r \geq 1$, a distance-$r$ dominating set (D$r$DS) 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 D$r$DSs $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of D$r$DSs 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). It is known that under any of $\mathsf{TS}$ and $\mathsf{TJ}$, the problem on split graphs is $\mathtt{PSPACE}$-complete for $r = 1$. We show that for $r \geq 2$, the problem is in $\mathtt{P}$, resulting in an interesting complexity dichotomy. Along the way, we prove some non-trivial bounds on the length of a shortest reconfiguration sequence on split graphs when $r = 2$ which may be of independent interest. Additionally, we design a linear-time algorithm under $\mathsf{TJ}$ on trees. On the negative side, we show that D$r$DSR for $r \geq 1$ on planar graphs of maximum degree three and bounded bandwidth is $\mathtt{PSPACE}$-complete, improving the degree bound of previously known results. We also show that the known $\mathtt{PSPACE}$-completeness results under $\mathsf{TS}$ and $\mathsf{TJ}$ for $r = 1$ on bipartite graphs and chordal graphs can be extended for $r \geq 2$.

Figures (11)

• Figure 1:
• Figure 2: The complexity status of D$r$DSR for fixed $r \geq 1$ on different graph classes under ${\mathsf{TJ}}$. Our results are for $r \geq 2$. Each arrow from graph class $A$ to graph class $B$ indicates that $B$ is properly included in $A$.
• Figure 3: Construction of a split graph $G = (K \uplus S, E)$ satisfying Lemma \ref{['lem:TS-split-shortest']}. Vertices in the light gray box are in $K$. Tokens in $D_s$ and $D_t$ are respectively marked by black and gray circles.
• Figure 4: Construction of a split graph $G = (K \uplus S, E)$ satisfying Lemma \ref{['lem:TJ-split-shortest']}. Vertices in the light gray box are in $K$. Tokens in $D_s$ and $D_t$ are respectively marked by black and gray circles.
• Figure 5: A tree $T_u$ rooted at $u = 1$. For $r = 2$, Algorithm \ref{['algo:minDrDS']} returns $D^\star = \{7, 5, 1\}$. A partition $\mathbb{P}(T_u) = \{C_7, C_5, C_1\}$ of $T_u$ is also constructed.

Theorems & Definitions (54)

• Proposition 3.1
• proof
• Proposition 3.2
• Corollary 3.3
• Proposition 3.4
• proof
• Corollary 3.5
• Lemma 3.6
• proof
• Corollary 3.7