A note on nongraphic matroids (Q801072)

The following well-known theorem is due to \textit{W. T. Tutte} [Trans. Am. Math. Soc. 90, 527-552 (1959; Zbl 0084.395)]: A matroid M is graphic if and only if it has no minor isomorphic to \(U^ 2_ 4\), \(F_ 7\), \(F^*_ 7\), \({\mathcal M}^*(K_ 5)\) or \({\mathcal M}^*(K_{3,3})\). Recall that \(U^ 2_ 4\) denotes the four-element matroid in which any 2-set is a base, \(F_ 7\) is the Fano matroid, \(F^*_ 7\) is its dual, \(K_ 5\) and \(K_{3,3}\) are the Kuratowski graphs, and \({\mathcal M}^*(K_ 5)\) or \({\mathcal M}^*(K_{3,3})\) are their bond matroids, respectively. In the paper under review, the authors prove the following variant of the above- mentioned Tutte's result: Let \(\{\) e,f,g\(\}\) be a circuit of a 3- connected nongraphic matroid M. Then M has a minor N isomorphic to one of \(U^ 2_ 4\), \(F_ 7\), \({\mathcal M}^*(K_{3,3})\) such that e,f,g\(\in E(N)\).