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 epsilon 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 epsilon V(D). The strong radius and strong diameter of D are defined as the minimum and maximum strong eccentricities se(u) among all u epsilon 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 x H) and sdiam(G x 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. (C) 2008 Elsevier B.V. All rights reserved.[[note]]SC

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 epsilon 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 epsilon V(D). The strong radius and strong diameter of D are defined as the minimum and maximum strong eccentricities se(u) among all u epsilon 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 x H) and sdiam(G x 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. (C) 2008 Elsevier B.V. All rights reserved.[[note]]SC

The optimal strong radius and optimal strong diameter of the Cartesian product graphs

NSFC [10831001]; Fujian Provincial Department of Education [JA10244]Let D be a strong digraph. The strong distance between two vertices u and v in D. denoted by sd(D)(u. v), is the minimum size (the number of arcs) of a strong subdigraph of D containing u and v. For a vertex v of D, the strong eccentricity se(u) is the strong distance between v and a vertex farthest from v. The minimum strong eccentricity among all vertices of D is the strong radius, denoted by srad(D), and the maximum strong eccentricity is the strong diameter, denoted by sdiam(D). The optimal strong radius (resp. strong diameter) srad(G) (resp. sdiam(G)) of a graph G is the minimum strong radius (resp. strong diameter) over all strong orientations of G. Juan et al. (2008) [Justie Su-Tzu Juan. Chun-Ming Huang, I-Fan Sun, The strong distance problem on the Cartesian product of graphs. Inform. Process. Lett. 107 (2008) 45-51] provided an upper and a lower bound for the optimal strong radius (resp. strong diameter) of the Cartesian products of any two connected graphs. In this work, we determine the exact value of the optimal strong radius of the Cartesian products of two connected graphs and a new upper bound for the optimal strong diameter. Furthermore, these results are also generalized to the Cartesian products of any n (n > 2) connected graphs. (C) 2010 Elsevier Ltd. All rights reserved