Obstructions to a binary matroid being graphic

Bixby and Cunningham showed that a 3-connected binary matroid MM is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid MM is graphic if and only if it has exactly r(M)+1r(M)+1 non-separating cocircuits. Hence the presence in MM of either an element in at least three non-separating cocircuits, or of at least r(M)+2r(M)+2 non-separating cocircuits, implies that MM is non-graphic. We provide lower bounds on the size of the set of such elements, and on the number of non-separating cocircuits, in such non-graphic binary matroids. A computationally efficient method for finding such lower bounds for specific minor-closed classes of matroids is given. Applications of this method and other results on sets of obstructions to a binary matroid being graphic are given.