# Infinitesimal rigidity in normed planes

## TL;DR

This paper characterizes infinitesimal rigidity of graphs in non-Euclidean normed planes using combinatorial and geometric methods, extending rigidity theory beyond Euclidean spaces.

## Contribution

It provides a complete combinatorial characterization of infinitesimal rigidity in non-Euclidean normed planes, using generalized Henneberg moves and properties of the unit ball.

## Key findings

- A graph has an infinitesimally rigid placement iff it contains a (2,2)-tight spanning subgraph.
- Rigid placements for K4 are constructed considering smoothness and convexity of the unit ball.
- The method extends rigidity theory to non-Euclidean geometries.

## Abstract

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 generalised 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.