The strong distance problem on the Cartesian product of graphs

Given a graph G and a strong oriented graph D=(V,F) of G. For u,v∈V(D), the strong distance sd(u,v) is the minimum size (the number of edges) of strong subdigraph of D containing u and v and the strong eccentricity se(u) is the maximum strong distance sd(u,v) for all v∈V(D). The strong radius and strong diameter of D are defined as the minimum and maximum strong eccentricities se(u) among all u∈V(D), respectively. The following four values: srad(G), SRAD(G), sdiam(G), and SDIAM(G) for a connected undirected graph G denote the minimum strong radius, maximum strong radius, minimum strong diameter, and maximum strong diameter, respectively, of D among all possible strong oriented graphs D of G.In this paper, we propose some important properties about these values and derive an inequality that gives an upper bound for each of srad(G×H) and sdiam(G×H) where G and H are any two graphs. Moreover, we can obtain srad(G) and sdiam(G) by substituting G into hypercube BCn or two extension hypercubes ECn and FCn, respectively. For these graphs, we also give a lower bound of SDIAM(G) for each of them.