On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating

TL;DR

This paper explores the relationships between dominating graphs, median graphs, and partial cubes, establishing equivalences and characterizations that connect domination properties with specific graph classes.

Contribution

It proves new equivalences linking dominating graphs with median graphs and minimal dominating sets, and characterizes when dominating graphs are median graphs or partial cubes.

Findings

Dominating graph of any graph is a partial cube.

Characterization of graphs where the dominating graph is a median graph.

Examples showing not all median graphs or partial cubes are dominating graphs.

Abstract

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are equivalent for every graph $G \not \simeq C_4$ with no isolated vertex, and in particular, that the simple third condition completely characterizes first two ones in which three concepts of dominating graphs, median graphs and complement of minimal dominating sets get related: - The dominating graph of G is a median graph, - The complement of every minimal dominating set of G is a minimal dominating set, - Every vertex of G is either of degree 1 or adjacent to a vertex of degree 1. As another result, we prove that the dominating graph of every graph is a partial cube and also give some examples to show that not all partial cubes or median graphs are isomorphic to the dominating graph of a graph. The above-mentioned results, as another highlight of the paper, provide novel infinite sources of examples of median graphs and partial cubes.