A constructive characterisation of circuits in the (2,1)-Sparse Matroid

A simple graph G = (V,E) is a (2,1)-circuit if |E| = 2|V| and |E(H)|<2|V(H)| for every proper subgraph H of G. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a constructive characterisation of (2,1)-circuits. The characterisation uses the well known 1-extension and X-replacement operations as well as several summation moves to glue together (2,1)-circuits over small cutsets.

This is joint work with Tony Nixon.