A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

Abstract

A simple graph G=(V,E) is a (2, 1)‐circuit if |E|=2|V| and |E(H)|≤2|V(H)|−1 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.