Maximum order of planar digraphs

Abstract

We consider the degree/diameter problem for directed planar graphs. We show that planar digraphs with diameter 2 and maximum out-degree and in-degree d, d ≥ 41, cannot have more than 2d vertices. We show that 2d is the best possible upper bound by constructing planar digraphs of diameter 2 having exactly 2d vertices. Furthermore, we give upper and lower bounds for the largest possible order of planar digraphs with diameter greater than 2. © Springer-Verlag Berlin Heidelberg 2005.