Strongly self-centered orientation of complete kk-partite graphs

For two vertices uu and vv in a strong oriented graph DD, the strong distance sd(u,v)sd(u,v) between uu and vv is the minimum size (the number of arcs) of a strong sub-digraph of DD containing uu and vv. For a vertex vv of DD, the strong eccentricity se(v)se(v) is the strong distance between vv and a vertex farthest from vv. The strong radius srad(D)srad(D) is the minimum strong eccentricity among the vertices of DD, and the strong diameter sdiam(D)sdiam(D) is the maximum strong eccentricity among the vertices of DD. An orientation DD of a graph GG is said to be a strongly self-centered orientation of GG if srad(D)=sdiam(D)srad(D)=sdiam(D). In this paper, we obtain some conditions for complete kk-partite graphs to have strongly self-centered orientations. Our results generalize a result on tournaments in Chartrand et al. (1999).