A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid

TL;DR

This paper offers a constructive way to identify circuits in the simple (2,2)-sparsity matroid, introducing new join operations and applying the results to frameworks on surfaces like the infinite circular cylinder.

Contribution

It introduces three join operations to fully characterize circuits in the simple (2,2)-sparsity matroid, extending the classical Henneberg approach.

Findings

Characterization of circuits using new join operations.

Extension to connected sparsity matroids.

Application to frameworks on surfaces, including the infinite circular cylinder.

Abstract

We provide a constructive characterisation of circuits in the simple (2,2)-sparsity matroid. A circuit is a simple graph G=(V,E) with |E|=2|V|-1 and the number of edges induced by any $X \subsetneq V$ is at most 2|X|-2. Insisting on simplicity results in the Henneberg operation being enough only when the graph is sufficiently connected. Thus we introduce 3 different join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces to provide a conjectured characterisation of when frameworks on an infinite circular cylinder are generically globally rigid.