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

Almost balanced biased graph representations of frame matroids