On the secure equitable domination in graphs

Abstract

A secure equitable dominating set S of a graph G is a dominating set in which for any vertex v ∈ V (G) \ S there exists at least one vertex u ∈ S such that u ∈ Ne(v), where Ne(v) indicate the equitable neighbourhood of v, and if we swap the vertex u with v, the equitable domination property of the graph will be unharmed. γᵉ sec(G) represents the secure equitable domination number of G, which is the cardinality of the minimum secure equitable dominating set in G. The improved bounds of the secure equitable domination number of some fundamental kinds of graphs are established in this study. Furthermore, we incorporate specific results based on the diameter, girth, and degree. Additionally, we determine the bounds of the secure equitable domination number of specific special classes of graphs.