Almost balanced biased graph representations of frame matroids

Abstract

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