Further results on the outer connected geodetic number of a graph

For a connected graph G of order at least two, an outer connected geodetic set S in a connected graph G is called a minimal outer connected geodetic set if no proper subset of S is an outer connected geodetic set of G. The upper outer connected geodetic number g+ oc(G) of G is the maximum cardinality of a minimal outer connected geodetic set of G. We determine bounds for it and find the upper outer connected geodetic number of some standard graphs. Some realization results on the upper outer connected geodetic number of a graph are studied. The proposed method can be extended to the identification of beacon vertices towards the network fault-tolerant in wireless local access network communication. Also, another parameter forcing outer connected geodetic number fog(G) of a graph G is introduced and several interesting results and realization theorem are proved.