Generalized Splitting and element splitting operations on $p$-matroids

TL;DR

This paper introduces generalized splitting and element splitting operations on $p$-matroids, characterizes their circuits and bases, and explores how these operations affect connectivity and Eulerian properties.

Contribution

It defines new splitting operations on $p$-matroids, characterizes their effects on circuits and bases, and analyzes connectivity and Eulerian properties under these operations.

Findings

Connectivity is preserved under element splitting.

A class of n-connected $p$-matroids is characterized.

Conditions for Eulerian $p$-matroids under splitting are provided.

Abstract

In this paper, we define generalized splitting and element splitting operations on $p$-matroids. $p$-matroids are the matroids representable over $GF(p).$ The circuits and the bases of the new matroid are characterized in terms of circuits and bases of the original matroid, respectively. A class of $n$-connected $p$-matroids which gives n-connected $p$- matroids using the generalized splitting operation is also characterized. We also prove that connectivity of $p$-matroid is preserved under element splitting operation. Sufficient conditions to obtain Eulerian $p$-matroid from Eulerian $p$-matroid under splitting and element splitting operations are provided.