The Geodetic Numbers of Cartesian Products of Graphs
Yongsheng Ye · Mathematica Applicata · 2007
For any two vertices u and v in a graph G,a u-v geodesic is the shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S)denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).In this paper,a sufficient and necessary condition for the equality of g(G)and g(G×K_3)is presented,and for a tree T,we give the geodetic number of T×K_m and C_n×K_m.