On extension of graphic matroids

On k-Connected Γ-Extensions of Binary Matroids

“…Let M be a binary matroid with ground set S and standard matrix representation A over GF (2). Let X = {x 1 , x 2 , .…”

“…Let M be a binary matroid with standard matrix representation A over GF (2) and let Y be a non-empty set of elements of M. Let A Y be the matrix obtained by adjoining one extra row to the matrix A whose entries are 1 in the columns labeled by the elements of the set Y and zero otherwise. The vector matroid of the matrix A Y , denoted by M Y , is called as the splitting matroid of M with respect to Y, and the transition from M to M Y is called as the splitting operation with respect to Y.…”

“…Azanchiler [1] extended this operation to binary matroids as follows: Definition 1.1. [1] Let M be a binary matroid with ground set S and standard matrix representation A over GF (2). Let X = {x 1 , x 2 , .…”

“…Note that the ground set of the matroid M X is S ∪ Γ and M X \ Γ = M. Therefore M X is an extension of M. The Γ-extension operation is related to the splitting operation on binary matroids, which is defined by Shikare et al [11], as follows: Definition 1.2. [11] Let M be a binary matroid with standard matrix representation A over GF (2) and let Y be a non-empty set of elements of M. Let A Y be the matrix obtained by adjoining one extra row to the matrix A whose entries are 1 in the columns labeled by the elements of the set Y and zero otherwise. The vector matroid of the matrix A Y , denoted by M Y , is called as the splitting matroid of M with respect to Y, and the transition from M to M Y is called as the splitting operation with respect to Y.…”

“…Azanchiler [1] characterized the circuits and the bases of the Γ-extension matroid M X in terms the circuits and bases of M, respectively. Some results on preserving graphicness of M under the Γ-extension operation are obtained in [2]. Borse and Mundhe [6] characterized the binary matroids M for which M X is graphic for any independent set X of M.…”