The lower and upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u-v path of G, where u, v is an element of D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori.