On the circuits of splitting matroids representable over GF(p)

TL;DR

This paper generalizes the splitting operation from binary to GF(p)-representable matroids, characterizes their structural properties, and explores conditions for Eulerian and connected matroids under this operation.

Contribution

It extends the splitting operation to p-matroids, providing new characterizations of circuits, bases, and conditions for Eulerian and connected matroids.

Findings

Characterization of circuits, bases, and independent sets of split p-matroids.

Conditions for Eulerian p-matroids after splitting.

Identification of connected p-matroids preserved under splitting.

Abstract

We extend the splitting operation from binary matroids (Raghunathan et al., 1998) to $p$- matroids, where $p$-matroids refer to matroids representable over $GF(p).$ We also characterize circuits, bases, and independent sets of the resulting matroid. Sufficient conditions to yield Eulerian $p$-matroids from Eulerian and non-Eulerian $p$-matroids by applying the splitting operation are obtained. A class of connected $p$-matroids that gives connected $p$-matroids under the splitting operation is characterized.