In this paper, we characterize the n-line splitting operation of graphs in terms of cycles of respective graphs and then extend this operation to binary matroids.In matroids, we call this operation as element-set splitting. For convenience, we call the resulting matroid, es-splitting matroid. We characterize circuits of es-splitting matroid. We also characterize the es-splitting matroid in terms of matrices.We also show that if M is a connected binary matroid then es-splitting matroid M e X is also connected.

We extend the notion of a point-addition operation from graphs to binary matroids. This operation can be expressed in terms of element-addition operation and splitting operation. We consider a special case of this construction and study its properties. We call the resulting matroid of this special case a Γ-extension of the given matroid. We characterize circuits and bases of the resulting matroids and explore the effect of this operation on the connectivity of the matroids.

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For a set  of circuits of a matroid M, G(B(M), ) is defined by the graph with one vertex for each basis of M, in which two basis B 1 and B 2 are adjacent if B 1 ∪ B 2 contains exactly one circuit and this circuit lies in . For two elements of a and b of ground set of a binary matroid M a splitting matroid M a,b is constructed. It is specified by two collections of circuits  0 and  1 dependent with collections of circuits of M. We want to study connectivity of G(B(M a,b),  0) and G(B(M a,b),  1).

In this article the author name Habib Azanchiler was incorrectly written as Habid Azanchiler.No other changes to the article have been made.