On minimal geodetic domination in graphs

Let G be a connected graph. For two vertices u and v in G, a u–v geodesic is any shortest path joining u and v. The closed geodetic interval IG[u, v] consists of all vertices of G lying on any u–v geodesic. For S ⊆ V (G), S is a geodetic set in G if ⋃ u,v∈S IG[u, v] = V (G). Vertices u and v of G are neighbors if u and v are adjacent. The closed neighborhood NG[v] of vertex v consists of v and all neighbors of v. For S ⊆ V (G), S is a dominating set in G if ⋃ u∈S NG[u] = V (G). A geodetic dominating set in G is any geodetic set in G which is at the same time a dominating set in G. A geodetic dominating set in G is a minimal geodetic dominating set if it does not have a proper subset which is itself a geodetic dominating set in G. The maximum cardinality of a minimal geodetic dominating set in G is the upper geodetic domination number of G. This paper initiates the study of minimal geodetic dominating sets and upper geodetic domination numbers of connected graphs.