Decomposition of Binary Signed-Graphic Matroids

TL;DR

This paper introduces a new decomposition theorem for binary signed-graphic matroids using Tutte's bridge theory, focusing on cocircuit deletion rather than traditional k-sum methods.

Contribution

It presents a novel decomposition approach for binary signed-graphic matroids based on cocircuit deletion, differing from previous k-sum based methods.

Findings

Decomposition characterized by minors resulting from cocircuit deletion

Identifies conditions under which minors are graphic or signed-graphic

Provides a new perspective on matroid decomposition techniques

Abstract

In this paper we employ Tutte's theory of bridges to derive a decomposition theorem for binary matroids arising from signed graphs. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on $k$-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minors resulting from the deletion of a cocircuit of a binary matroid will be graphic matroids apart from exactly one that will be signed-graphic, if and only if the matroid is signed-graphic.