Characterizing Graphs via Edge Geodetic Domination Number

Dominance in graphs is the crucial aspect of graph theory that has been thoroughly examined. Consider a graph \(G_1(V_1,E_1)\) with \(S⊆ V_1\) in such a way that at least one vertex in the set is adjacent to the vertices that do not belong to the set then \(S\) is said to be the dominating set. In other words, it can be said that the set of vertices belonging to \(S\) and \(S'\) has at least a single neighbor in each other. Any set \(A⊆ G_1\) having all edges of \(G\) comprised in a geodesic uniting a pair of vertices in \(A\) is claimed to be an EG-set of \(G_1\). The EG-number, indicated by the symbol \(g_e(G)\), is the lowest order of its EG-set. A \(g_e\)-set of \(G\) or EG-basis of \(G\), is any EG-set of order \(g_e(G)\). If a collection of vertices \(D\) in \(G\) is together an EG-set and a dominant set then \(D\) is considered an EG-dominating set. The EG-dominance number of the EG-dominating set is its minimum cardinal number represented as \(γge(G)\). With this work, we explore the EG-dominance number of varied graphs namely antiprism graph \(A_n\), alternate pentagonal snake \(A(_n)\), Bistar graph, ladder graph, jewel graph and Helm graph.