The $k$-Dominating Graph

TL;DR

This paper introduces the concept of the $k$-dominating graph to analyze the reconfiguration of dominating sets in a graph, providing conditions for its connectivity which aids in understanding how dominating sets can be transformed.

Contribution

It defines the $k$-dominating graph and establishes conditions under which this graph is connected, advancing the study of dominating set reconfiguration.

Findings

Provides conditions for the connectivity of $D_k(G)$

Enhances understanding of dominating set reconfiguration pathways

Facilitates analysis of reconfiguration sequences in graphs

Abstract

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the corresponding dominating sets of $G$ differ by either adding or deleting a single vertex. The graph $D_k(G)$ aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of $D_k(G)$. In this paper we give conditions that ensure $D_k(G)$ is connected.