Almost balanced biased graph representations of frame matroids

TL;DR

This paper characterizes all biased graphs representing a specific 3-connected nongraphic frame matroid with a balancing vertex, showing a near-uniqueness of representation up to certain local modifications.

Contribution

It provides a complete description of biased graphs representing a given 3-connected nongraphic frame matroid with a balancing vertex, establishing near-uniqueness of representation.

Findings

All biased graphs with the same frame matroid are obtained by local modifications.

The representation of 4-connected nongraphic frame matroids with a balancing vertex is essentially unique.

Biased graphs representing the same matroid differ only by replacing edges incident to the balancing vertex with unbalanced loops.

Abstract

Given a 3-connected biased graph $\Omega$ with a balancing vertex, and with frame matroid $F(\Omega)$ nongraphic and 3-connected, we determine all biased graphs $\Omega'$ with $F(\Omega') = F(\Omega)$. As a consequence, we show that if $M$ is a 4-connected nongraphic frame matroid represented by a biased graph $\Omega$ having a balancing vertex, then $\Omega$ essentially uniquely represents $M$. More precisely, all biased graphs representing $M$ are obtained from $\Omega$ by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.