Characterizing Graphs via Edge Geodetic Domination Number - Arvind; Mehra, Seema (Communications in Mathematics and Applications, 2026)
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 \(S\subseteq V_1\) 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 \(A\subseteq G_1\) 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 \(γ_{ge}(G)\). With this work, we explore the EG-dominance number of varied graphs namely antiprism graph \(A_n\), alternate pentagonal snake \(A(\PS_n)\), Bistar graph, ladder graph, jewel graph and Helm graph.
Keywords: Edge geodetic dominating set; Edge geodetic dominance number; Antiprism graph alternate pentagonal snake graph; Bistar graph; Ladder graph