Extension of line-splitting operation from graphs to binary matroids

Abstract

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.

On the structure of spikes

“…For r ! 3, let A r be the r  ð2r þ 1Þ matrix ½I r jJ r À I r j1 over GF (2) whose columns are labeled, in order, x 1 , x 2 , :::, x r , y 1 , y 2 :::, y r , t: The vector matroid M½A r of this matrix is called the rank-r binary spike with tip t and denoted by Z r . Oxley [4] showed that all rank-r, 3-connected binary matroids without a 4-wheel minor can be obtained from a binary r-spike by deleting at most two elements.…”

“…In this section, our main goal is to give a necessary and sufficient condition for X EðZ r Þ with e 2 X, to obtain Z rþ1 by applying the es-splitting operation on X. Now suppose that M ¼ Z r be a binary rank-r spike with the matrix representation ½I r jJ r À I r j1 over GF (2) whose columns are labeled, in order x 1 , x 2 , :::, x r , y 1 , y 2 , :::, y r , t:…”

“…In the following lemma, we shall use the well-known facts that if a matroid M is n-connected with EðMÞ ! 2ðn À 1Þ, then all circuits and all cocircuits of M have at least n elements, and if A is a matrix that represents M over GF (2), then the cocircuit space of M equals the row space of A.…”

“…Azanchiler [1,2] extended the notion of n-line splitting operation from graphs to binary matroids. He characterized the n-line splitting operation of graphs in terms of cycles of the respective graph and then extended this operation to binary matroids as follows.…”

“…He characterized the n-line splitting operation of graphs in terms of cycles of the respective graph and then extended this operation to binary matroids as follows. Let M be a binary matroid on a set E and let X be a subset of E with e 2 X: Suppose A is a matrix that represents M over GF (2). Let A e X be a matrix obtained from A 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 labeled a and c to the resulting matrix such that the column labeled a is zero everywhere except in the last row where it takes the value 1, and c is the sum of the two column vectors corresponding to the elements a and e. The vector matroid of the matrix A e X is denoted by M e X : The transition from M to M e X is called an es-splitting operation.…”