Reconfiguration graphs for minimal domination sets

TL;DR

This paper introduces a reconfiguration graph model for minimal dominating sets in graphs, exploring its properties and connectivity in specific graph classes, and classifying graphs based on the structure of their reconfiguration graphs.

Contribution

It defines a new reconfiguration graph for minimal dominating sets, analyzes its connectivity in trees and split graphs, and classifies graphs with complete or empty reconfiguration graphs.

Findings

Reconfiguration graph is connected for trees and split graphs.

Classified graphs with complete reconfiguration graphs.

Classified graphs with empty reconfiguration graphs.

Abstract

A dominating set $S$ in a graph is a subset of vertices such that every vertex is either in $S$ or adjacent to a vertex in $S$. A minimal dominating set $M$ is a dominating set such that $M-v$ is not a dominating set for all $v \in M$. In this paper we introduce a reconfiguration graph $\mathcal{R}(G)$ for minimal dominating sets under a generalization of the token sliding model. We give some preliminary results which include showing that $\mathcal{R}(G)$ is connected for trees and split graphs. Additionally we classify all graphs which have $\mathcal{R}(G) = K_n$ and $\mathcal{R}(G) = \overline{K_n}$ for all $n$.