Biased graphs whose matroids are special binary matroids (Q1813718)

A biased graph \(\Omega\) is a graph \(\Gamma\) together with a class \({\mathcal B}\) of polygons of \(\Gamma\) such that no theta-subgraph of \(\Gamma\) contains exactly two members of \({\mathcal B}\). (Examples arise form signed graphs by letting \({\mathcal B}\) consist of the polygons with an even number of minus-signs.) A subgraph \(S\) is balanced if each polygon of \(S\) is in \({\mathcal B}\), contrabalanced if none does. The associated bias matroid \(G(\Omega)\) is defined on the edge set of \(\Gamma\) and has as circuits the balanced polygons and the minimal contrabalanced connected edge sets with cyclomatic number two. The lift \(L(\Omega)\) and the complete lift \(L_ 0(\Omega)\) are two more matroids closely related to \(G(\Omega)\). The article determines the possible structure of \(\Omega\) in the cases where one of the three assocaited matroids is known to be one of the matroids \(F_ 7\), \(R_{10}\), \(G(K_{3,3})\), \(G(K_ 4)\), \(G(K_ 5)\), and their duals. Sample result: \(L_ 0(\Omega) \cong F^*_ 7\) if and only if \(\Omega\) arises from \(K_ 4\) with all edges negatively signed. Moreover, \(L_ 0(\Omega)\) is regular if and only if \(\Omega\) is sign- biased and has no subgraph that is a subdivision of \([-K_ 4]\) or a cycle of tree unbalanced polygons.