Weighted digraphs and tropical cones

TL;DR

This paper explores the combinatorics of tropical hyperplane arrangements and their associated cell decompositions using convex polyhedra linked to weighted digraphs, solving a conjecture in the field.

Contribution

It introduces a novel approach connecting weighted digraphs with tropical hyperplane arrangements, advancing understanding of their combinatorial structure and resolving a conjecture.

Findings

Relation between tropical hyperplane arrangements and convex polyhedra

Natural cell decompositions of tropical projective space

Resolution of a conjecture by Develin and Yu (2007)

Abstract

This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of $\mathbb{R}^d$. This leads to natural cell decompositions of the tropical projective space $\mathbb{TP}_{\min}^{d-1}$. 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 results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007).