Lower and Upper Orientable Strong Radius and Strong Diameter of the Cartesian Product of Complete Graphs

Abstract

For two vertices \(u\) and \(v\) in a strong digraph \(D\), the strong distance \(sd(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). The strong eccentricity \(se(v)\) of a vertex \(v\) of \(D\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(srad(D)\) (resp. strong diameter \(sdiam(D)\)) of \(D\) is the minimum (resp. maximum) strong eccentricity among all vertices of \(D\). The lower (resp. upper) orientable strong radius \(srad(G)\) (resp. \(SRAD(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong radius over all strong orientations of \(G\). The lower (resp. upper) orientable strong diameter \(sdiam(G)\) (resp. \(SDIAM(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong diameter over all strong orientations of \(G\). In this paper, we determine the lower orientable strong radius and strong diameter of the Cartesian product of complete graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of the Cartesian product of complete graphs.