On Pancyclicity in a Mixed Model for Domination Reconfiguration

TL;DR

This paper introduces a new domination reconfiguration model combining TAR and TS rules, showing that for certain graph classes, the resulting graphs are pancyclic, and explores properties of the join operation.

Contribution

It demonstrates that adding a small number of token sliding edges can make the domination reconfiguration graph pancyclic for specific graph classes.

Findings

TARS-graphs of trees, complete graphs, and complete multipartite graphs are pancyclic.

The join of two graphs with pancyclic TARS-graphs is also pancyclic.

TARS-graphs of some classes can be made pancyclic by adding few edges.

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, we show that their TARS-graphs will be pancyclic. Notably, we prove that if the TARS-graphs of $G$ and $H$ are pancyclic, then the TARS-graph of the join $G \vee H$ will also be pancyclic. We conclude by posing the question: Are all TARS-graphs pancyclic?