Almost balanced biased graph representations of frame matroids

Abstract

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