Constructing internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2013 ElsevierIn an earlier paper, we proved that an internally 4-connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from the original. We also describe each as a bottom-up operation, constructing a larger matroid from the original, and we give necessary and su fficient conditions for each of these bottom-up moves to produce an internally 4-connected binary matroid. From this, we derive a constructive method for generating all internally 4-connected binary matroids.This study is supported by NSF IRFP Grant 0967050, the Marsden Fund, and the National Security Agency

Element splitting operation for matroids representable over GF(p)GF(p)

The element splitting operation on binary matroids due to Shikare, is a natural generalization of Slater's {\it n-point splitting operation} on graphs. The present paper generalizes the notion of {\it n-point splitting operation} on graphs to matroids representable over $GF(p)$. This operation on a given matroid representable over $GF(p)$ yields a matroid representable over $GF(p)$. We characterize circuits, bases and hyperplanes of the resulting matroid in terms of the circuits, bases and hyperplanes of the original matroid $M$, respectively. We also explore the effect of this operation on Eulerian, bipartite and connected matroids which are representable over $GF(p)$. This operation, in general, may not preserve the connectedness of the given matroid. We provide a necessary and sufficient condition for a connected matroid representable over $GF(p)$ to yield a connected matroid representable over $GF(p)$ by the element splitting operation