Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids

TL;DR

This paper introduces cobiased graphs, a new dual graphical structure, to characterize single-element extensions and quotients of graphic matroids, expanding the understanding of matroid modifications.

Contribution

The paper presents cobiased graphs as a novel dual structure to biased graphs, providing new characterizations of matroid extensions and quotients.

Findings

Cobiased graphs effectively characterize single-element extensions of graphic matroids.

They also describe elementary quotients of graphic matroids.

The work extends graphical matroid theory with a new dual structure.

Abstract

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.