🤖 AI Summary
This study investigates the reconfiguration problem of distance-$r$ dominating sets (D$r$DS) under Token Jumping (TJ) and Token Sliding (TS) rules, focusing on computational complexity for $r geq 2$ over split graphs, trees, planar graphs, bipartite graphs, and chordal graphs. Using graph-theoretic analysis, intricate reductions, and constructive algorithm design, we establish: (i) a phase transition in split graphs—from PSPACE-completeness for $r = 1$ to polynomial-time solvability for $r geq 2$; (ii) a polynomial-time algorithm for D$r$DS reconfiguration on split graphs when $r geq 2$; (iii) the first nontrivial upper bound on shortest reconfiguration sequence length for $r = 2$; (iv) a linear-time TJ algorithm for trees; and (v) PSPACE-completeness extensions to planar graphs of maximum degree 3, graphs of bounded bandwidth, bipartite graphs, and chordal graphs—for all $r geq 1$.
📝 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 $r geq 2$.
