On Pancyclicity in a Mixed Model for Domination Reconfiguration

TL;DR

The paper addresses the limitation that domination reconfiguration graphs under TAR, denoted $\mathcal{D}(G)$, rarely contain Hamilton cycles by introducing the TARS-graph $\varepsilon(G)$ that augments TAR adjacencies with a small number of TS adjacencies. It develops structural lifting lemmas (Operations A and B) and proves that $\varepsilon(T)$ is pancyclic for every tree $T$, then extends these results to graph joins $G \vee H$, showing that many graph families (including complete multipartite and complete split graphs) yield pancyclic TARS-graphs. The approach combines combinatorial constructions with Gray-code ideas to generate cycles of all lengths, establishing a strong cycle-rich structure in the reconfiguration space. The results raise open questions about the universality of pancyclicity in TARS-graphs and the minimal TS augmentation needed to achieve it, with potential implications for Gray-code-like enumerations of dominating sets. Overall, the work demonstrates that a modest augmentation of the TAR model can yield highly connected and cycle-rich reconfiguration graphs.

Abstract

A new model for domination reconfiguration is introduced which combines the properties of the preexisting token addition/removal (TAR) and token sliding (TS) models. The vertices of the TARS-graph correspond to the dominating sets of $G$, where two vertices are adjacent if and only if they are adjacent via either the TAR reconfiguration rule or the TS reconfiguration rule. While the domination reconfiguration graph obtained by using only the TAR rule (sometimes called the dominating graph) will never have a Hamilton cycle, we show that for some classes of graphs $G$, by adding a relatively small number of token sliding edges, the resulting graph is not only hamiltonian, but is in fact pancyclic. In particular, if the underlying graphs are trees, complete graphs, or complete multipartite graphs, their TARS-graphs will be pancyclic. We also provide pancyclicity results for TARS-graphs of graph unions and joins, and conclude by posing the question: Are all TARS-graphs pancyclic?

Figures (4)

• Figure 1: The TARS-graph of $K_{1,3}$ where the vertices in each dominating set are coloured red.
• Figure 2: The binary-reflected Gray code on $5$ bits, read column-wise.
• Figure 3: Operation A (left) and Operation B (right).
• Figure 4: A schematic diagram of the cycles used in the proof of Theorem \ref{['thm:joins']}.

Theorems & Definitions (15)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Proposition 3.1
• Corollary 3.2
• Corollary 3.3
• Corollary 3.4
• Lemma 3.5