A note on connectivity of splitting matroids

TL;DR

This paper investigates the conditions under which the splitting operation on disconnected binary matroids results in a connected matroid, extending the concept from graph theory to matroid theory.

Contribution

It provides a characterization of binary disconnected matroids whose splitting matroids are connected, filling a gap in the understanding of splitting operations in matroid theory.

Findings

Characterization of binary disconnected matroids with connected splitting matroids

Extension of splitting operation concepts from graphs to binary matroids

Insight into connectivity preservation under splitting in matroids

Abstract

Fleischner introduced the idea of splitting a vertex of degree at least three in a connected graph and used the operation to characterize Eulerian graphs. Raghunathan et. al. extended the splitting operation from graphs to binary matroids. It has been studied that splitting operation, in general, may not preserve the connectedness of the binary matroid. Interestingly, it is true that the splitting matroid of a disconnected matroid may be connected. In this paper, we characterize the binary disconnected matroids whose splitting matroid is connected.