On Pancyclicity in a Mixed Model for Domination Reconfiguration

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?