The strong distance problem on the Cartesian product of graphs

Abstract

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 BC n or two extension hypercubes EC n and FC n , respectively. For these graphs, we also give a lower bound of SDIAM ( G ) for each of them.