A generalization of a graph result of D. W. Hall
In 1943, D. W. Hall proved that if a simple 3-connected graph has \(K_5\) as a minor, then either it has \(K_{3,3}\) as a minor, or it equals \(K_5\). The author generalizes this result to binary matroids. In particular, she proves that if a 3-connected binary matroid has \(M(K_5)\) as a minor, then either it has \(M(K_{3,3})\) or its dual as a minor, or it equals one of \(M(K_5)\), \(T_{12}\), or a single-element contraction of \(T_{12}\). The matroid \(T_{12}\) is a 12-element rank-6 self-dual matroid with a transitive automorphism group. Moreover, \(T_{12}\) is represented over GF(2) by the \(15 \times 12\) matrix whose rows are indexed by the edges of the Petersen graph \(P_{10}\) and whose columns are the incidence vectors of the 5-cycles of \(P_{10}\).
scientific article; zbMATH DE number 1062445