Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling

Abstract

Claude Berge (1987) introduced the concept of k-extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph (Formula presented.). Graphs possessing the property of being 1-extendable are termedas Berge graphs. This introduction gave rise to the notion of well-covered graphs and well-dominated graphs. A graph is categorized as well-covered if each of its maximal independent sets is, in fact, a maximum independent set. Similarly, a graph attains the classification of well-dominated if every minimal dominating set (DS) within it is a minimum dominating set. In alignment with the concept of k-extendable graphs, the framework of (Formula presented.) -endowed graphs and symmetric (Formula presented.) -endowed graphs are established. In these graphs, each DS of size k encompasses a minimum DS of the graph. In this article, a study of (Formula presented.) -endowed dominating sets is initiated. Various results providing a deep insight into (Formula presented.) -endowed dominating sets in graphs such as those characterizing the ones possessing a unique minimum DS are proven. We also introduce and study the symmetric (Formula presented.) -endowed graphs and minimality of dominating sets in them. In addition, we give a solution to an open problem in the literature. which seeks to find a domination-based parameter that has a correlation coefficient of (Formula presented.) with the total (Formula presented.) -electronic energy of lower benzenoid hydrocarbons. We show that the upper dominating number (Formula presented.) studied in this paper delivers a strong prediction potential.