Graphic Elementary Lift of Cographic Matroids

TL;DR

This paper investigates the conditions under which elementary lifts of cographic matroids result in graphic matroids, contributing to the understanding of matroid transformations and their properties.

Contribution

It characterizes cographic matroids whose elementary lifts are graphic, advancing the theory of matroid lifts and their structural implications.

Findings

Identifies conditions for elementary lifts of cographic matroids to be graphic

Provides characterization of cographic matroids with graphic elementary lifts

Enhances understanding of matroid lift operations and their effects

Abstract

A matroid $N$ is a lift of a binary matroid $M$, if $N=Q\backslash X$ when $Q/X=M$ for some binary matroid $Q$ and $X \subseteq E(Q)$ and is called an elementary lift of $M$, if $|X|=1$. A splitting operation on a binary matroid can result in an elementary lift. An elementary lift of a cographic or a graphic matroid need not be cographic or graphic. We intend to characterize the cographic matroids whose elementary lift is a graphic matroid.