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. We further show that our approach also allows for computing Chow Betti numbers for complete toric varieties.

Abstract. We characterize rigid graphs for cylindrical normed spaces [Formula: see text] where [Formula: see text] is a finite-dimensional real normed linear space and [Formula: see text] is endowed with the product norm. In particular, we obtain purely combinatorial characterizations of minimal rigidity for a large class of 3-dimensional cylindrical normed spaces; for example, when [Formula: see text] is an [Formula: see text]-plane with [Formula: see text]. We also characterize rigid graphs in the 4-dimensional cylindrical space [Formula: see text]. These are among the first combinatorial characterizations of rigid graphs in normed spaces of dimension [Formula: see text].

Abstract We present three results which support the conjecture that a graph is minimally rigid in d-dimensional $$\ell _p$$ ℓ p -space, where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$p\not =2$$ p ≠ 2 , if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $$\ell _p^d$$ ℓ p d to $$\ell _p^{d+1}$$ ℓ p d + 1 . We then prove that every (d, d)-sparse graph with minimum degree at most $$d+1$$ d + 1 and maximum degree at most $$d+2$$ d + 2 is independent in $$\ell _p^d$$ ℓ p d . Finally, we prove that every triangulation of the projective plane is minimally rigid in $$\ell _p^3$$ ℓ p 3 . A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.

Abstract We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth’s 1978/1979 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we obtain upper bounds for the dimension of the space of trivial motions for a framework and establish the flexibility of small frameworks in general non-Euclidean normed spaces.

We present a piece of software for computing animations of linkages with rotational symmetry in the plane. We construct these linkages from an algorithm that utilises a special type of edge colouring to embed graphs with rotational symmetry.

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a (2,2)-tight spanning subgraph. The method uses an inductive construction based on generalized Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.