Almost balanced biased graph representations of frame matroids
Abstract
Given a 3connected biased graph $\Omega$ with a balancing vertex, and with frame matroid $F(\Omega)$ nongraphic and 3connected, we determine all biased graphs $\Omega'$ with $F(\Omega') = F(\Omega)$. As a consequence, we show that if $M$ is a 4connected 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.07370
 Bibcode:
 2016arXiv160607370D
 Keywords:

 Mathematics  Combinatorics