On geodomination in graphs (Q2717916)

Let \(G=(V,E)\) be a nontrivial connected graph. A set \(S\) of vertices of \(G\) is called geodominating if every vertex of \(G\) lies on a shortest path between two vertices in \(S\). A set \(S\) of vertices of \(G\) is called open geodominating if every vertex \(u\) of \(G\) either lies on a shortest path between two vertices in \(S\) which are different from \(u\) or the neighbourhood \(N(u)\) of \(u\) induces a complete graph and \(u\in S\). The (open) geodomination number \(g(G)\) (\(og(G)\)) of \(G\) is the minimum cardinality of an (open) geodominating set of \(G\). The authors study the behaviour of \(g\) under the addition of a pending vertex to the graph. Furthermore, they obtain some results for the geodomination number of Cartesian products of graphs and determine the open geodomination number for some special classes of graphs.