Splitting in a binary matroid

Some Results Involving the Splitting Operation on Binary Matroids

“…Consider the non-adjacent elements 1 and 6 of M. Then, by Lemma 1.2, M 1,6 S, C 1,6 is the matroid with ground set S and circuit set C 1,6 . By contracting 1 in M 1,6 , we get a matroid M 1,6 /{1}, which is isomorphic to the matroid M * K 3,3 .…”

“…Raghunathan et al 6 extended the splitting operation from graphs to binary matroids as follows. Definition 1.1.…”

Splitting Lemma for 2-Connected Graphs

“…For matroid concepts and terminology, we refer to Oxley 2 . Raghunathan et al 3 generalized the splitting operation of graphs to binary matroids and characterized Eulerian matroids in terms of this operation. We characterize connected Eulerian graphic matroids.…”

Extension of line-splitting operation from graphs to binary matroids

“…In [2] Raghunathan, Shikare and Waphare generalized the splitting operation of graphs to binary matroids. Shikare, Azadi and Waphare [6,7] extended the notions of n-point splitting from graphs to binary matroids.…”

“…Let M be a binary matroid on a set S and X be a subset of Se ∈ X. Suppose that A is a matrix over GF (2), that represents the matroid M. Let A e X be the matrix that is obtained by adjoining an extra row to A with this row being zero everywhere except, in the columns corresponding to the elements of X where it, takes the value 1 and then adjoining two columns a and γ to the resulting matrix such that the column a is zero everywhere except in the last row (new row) where it takes the value 1, and γ is a sum of two column vectors corresponding to a and e.…”