  Extremality and irreducibility constitute fundamental concepts in
mathematics, particularly within tropical geometry. While extremal
decomposition is typically computationally hard, this article presents a fast
algorithm for identifying the extremal decomposition of tropical varieties with
rational balanced weightings. Additionally, we explore connections and
applications related to rigidity theory. In particular, we prove that a
tropical hypersurface is extremal if and only if it has a unique reciprocal
diagram up to homothety.
