The Lower And Upper Forcing Geodetic Numbers Of C_m×C_n×C_k

A set D of vertices of a graph G is called a geodetic set if each vertex of G lies on some shortest u-v path of G,where u,v∈D.A geodetic set with the minimum cardinality is called a minimum geodetic set(g-set for short) and its cardinality is called the geodetic number of G.A subset S of a g-set D is called a forcing subset of D if D is the unique g-set containing S.The forcing geodetic number of D is the minimum cardinality of a forcing subset of D,and the lower and upper forcing geodetic number of a graph G are the minimum and maximum forcing geodetic numbers,respectively, among all g-sets of G.In this paper,we use the property of the Cartesian product to discuss the structures of g-sets of C_m×C_n×C_k,and find out the geodetic numbers, the lower and upper forcing geodetic numbers of C_m×C_n×C_k.