Dominating Sets in Graph Theory and Algebraic Hyperstructures

Abstract

The study of hyperstructures and their connections with other branches of mathematics has been explored by various researchers. In this context, several studies have investigated the relationship between hyperstructures and Graph Theory. This paper aims to establish new connections between Hyperstructure Theory and Graph Theory by focusing on the concept of
dominating sets and minimal dominating sets of a graph. Specifically, we define different semihypergroups on the set of all dominating sets and the set of all minimal dominating sets of a given graph. We also examine the conditions under which some of these semihypergroups can be hypergroups, and provide examples to illustrate them. Finally, we present some theorems that introduce and construct numerous graphs in which some of these semihypergroups are hypergroups.