On the $n-$dominating graph of specific graphs

TL;DR

This paper investigates the structure and properties of the $n$-dominating graph for specific classes of graphs, providing insights into how dominating sets relate within these graph constructs.

Contribution

It introduces the concept of the $n$-dominating graph for specific graphs and analyzes its properties, expanding understanding of dominating set relationships.

Findings

Characterization of $n$-dominating graphs for certain graph classes

Identification of structural properties of $D_n(G)$

Insights into the adjacency relations among dominating sets

Abstract

Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. 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. In this paper we consider and study the $n$-dominating graph of specific graphs.