The tape reconfiguration problem and its consequences for dominating set reconfiguration

TL;DR

This paper studies the reconfiguration of dominating sets under token sliding by introducing Tape-Rec, a multi-tape reconfiguration problem that serves as a backbone for transferring hardness to Dominating Set Reconfiguration (DSR). It establishes an explicit hardness threshold for TS-DSR using a novel Tape-Rec construction, and shows a fundamental separation between token sliding and token jumping on bounded-treewidth graphs, along with XL-hardness results parameterized by solution size plus graph structure such as feedback vertex set. On the positive side, the authors prove that TS-DSR is FPT on planar graphs and several sparse graph classes (e.g., $K_{3,d}$-free and $K_{4,d}$-minor-free), expanding the landscape beyond token jumping variants. The work leverages domination-core techniques and a structured reduction from Tape-Rec to DS-reconfiguration, providing a new methodological lens for understanding reconfiguration problems and highlighting important gaps between sliding and jumping models in graph classes with restricted width.

Abstract

A dominating set of a graph $G=(V,E)$ is a set of vertices $D \subseteq V$ whose closed neighborhood is $V$, i.e., $N[D]=V$. We view a dominating set as a collection of tokens placed on the vertices of $D$. In the token sliding variant of the Dominating Set Reconfiguration problem (TS-DSR), we seek to transform a source dominating set into a target dominating set in $G$ by sliding tokens along edges, and while maintaining a dominating set all along the transformation. TS-DSR is known to be PSPACE-complete even restricted to graphs of pathwidth $w$, for some non-explicit constant $w$ and to be XL-complete parameterized by the size $k$ of the solution. The first contribution of this article consists in using a novel approach to provide the first explicit constant for which the TS-DSR problem is PSPACE-complete, a question that was left open in the literature. From a parameterized complexity perspective, the token jumping variant of DSR, i.e., where tokens can jump to arbitrary vertices, is known to be FPT when parameterized by the size of the dominating sets on nowhere dense classes of graphs. But, in contrast, no non-trivial result was known about TS-DSR. We prove that DSR is actually much harder in the sliding model since it is XL-complete when restricted to bounded pathwidth graphs and even when parameterized by $k$ plus the feedback vertex set number of the graph. This gives, for the first time, a difference of behavior between the complexity under token sliding and token jumping for some problem on graphs of bounded treewidth. All our results are obtained using a brand new method, based on the hardness of the so-called Tape Reconfiguration problem, a problem we believe to be of independent interest.

Figures (2)

• Figure 1: The instance of Sync-Multi-Tape-Rec equivalent to the instance of Dominating Set for the $5$-cycle and where $k = 2$.
• Figure 2: The construction from Theorem \ref{['thm:auto_tw4']}, and the corresponding moves of the reading heads encoding the move of the token $\tau_2$ from $4$ to $5$. Here $V_1=\{1,2,3\}$ and $V_2=\{4,5\}$.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Remark 9
• Theorem 10