Weighted digraphs and tropical cones

This paper is about the combinatorics of finite point con-figurations in the tropical projective space or, dually, of ar-rangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be con-sidered via coarsenings of the resulting polyhedral decompo-sitions of ℝ^d. This leads to natural cell decompositions of the tropical projective space TP^d-1min. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use re-sults from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007).