Reconfiguration graphs for minimal domination sets

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 $n$.

Figures (4)

• Figure 1: The two minimal dominating sets of $K_{1,n}$
• Figure 2: 3 by 3 rook's graph and folded rook's graphs
• Figure 3: Examples of folded and altered folded rook's graph
• Figure 4: A graph with disconnected $\mathcal{R}(G)$

Theorems & Definitions (39)

• Definition 1.1: Expansion/Contraction Model
• Theorem 1.3
• Theorem 2.1
• proof
• Corollary 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5