Generalization of Splitting off Operation to Binary Matroids

The splitting off operation for graphs is defined in the following way: Let G be a graph. Given incident edges x = vv1 and y = vv2 in G, we can construct a new graph Gxy by adding the edge v1v2 and deleting the edges x and y. If v1 = v2, then the resulting loop is deleted. The transition from G to Gxy is called the splitting off operation. litting off operation has important applications in graph theory (for example, see Frank A., Augmenting graphs to meet edge-connectivity requirements, SIAM J. of Discrete Mathematics 5 No.1. (1992), 22-53; Jordan T., Edge-Splitting Problems With Demands, Springer LINK: Lecture Notes in Computer Science 1610, (1999); and Lovasz L., Combinatorial Problems and Exercises, North Holland, Amsterdam (1979)). racterize the circuits of the graph Gxy in terms of the circuits of G.