(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G be a simple graph. A subset D of vertices in G is a dominating set of G if every vertex not in D has at least one neighbor in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. An edge geodetic set of G is a set S V(G) such that every edge of G is contained in a geodetic joining some pair of vertices in S. The edge geodetic number g_e(G) of G is the minimum cardinality of edge geodetic set. A set of vertices S in G is an edge geodetic dominating set of G if S is both an edge geodetic set and a dominating set. The minimum cardinality of an edge geodetic dominating set of G is its edge geodetic domination number and is denoted by γ_g_e(G). In this study, we determined the edge geodetic domination number of graphs obtained through the deletion of independent edges of complete graphs and graphs resulting from the K_r-gluing of complete graphs. It is also shown that for any positive integers 2 ≤ a ≤ b, there exists a connected graph G such that g_e(G)=a and γ_g_e(G)=b.