Directed hypergraphs and applications
Discrete Applied Mathematics - Special issue: combinatorial structures and algorithms
Tropical convexity via cellular resolutions
Journal of Algebraic Combinatorics: An International Journal
The number of extreme points of tropical polyhedra
Journal of Combinatorial Theory Series A
Minimal half-spaces and external representation of tropical polyhedra
Journal of Algebraic Combinatorics: An International Journal
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Tropical polyhedra are known to be representable externally, as intersections of finitely many tropical half-spaces. However, unlike in the classical case, the extreme rays of their polar cones provide external representations containing in general superfluous half-spaces. In this paper, we prove that any tropical polyhedral cone in R^n (also known as ''tropical polytope'' in the literature) admits an essentially unique minimal external representation. The result is obtained by establishing a (partial) anti-exchange property of half-spaces. Moreover, we show that the apices of the half-spaces appearing in such non-redundant external representations are vertices of the cell complex associated with the polyhedral cone. We also establish a necessary condition for a vertex of this cell complex to be the apex of a non-redundant half-space. It is shown that this condition is sufficient for a dense class of polyhedral cones having ''generic extremities''.