Recognizing signed-graphic matroids: Cylinder flips and the importance of column scaling

TL;DR

This paper explores the role of column scaling in relating different signed-graphic representations of the same matroid, introducing the cylinder flip operation and analyzing their interconnections using computational examples.

Contribution

It demonstrates the necessity of column scaling for transforming signed-graphic representations and introduces the cylinder flip operation as a new matroid-preserving transformation.

Findings

Column scaling is sometimes essential for representation transformation.

Multiple signed-graphic representations can row-reduce to the same form.

The cylinder flip relates certain pairs of signed-graphic representations.

Abstract

In this paper, we investigate the importance of column scaling in relating two signed-graphic representations of the same matroid. We used the Sage Mathematics software to generate many examples of signed-graphic matroids and their signed-graphic representations. Our examples show that column scaling is sometimes necessary in order to transform one signed-graphic representation into another; moreover, there exist many collections of signed-graphic representations that row-reduce to the same standard form. We also discuss an interesting matroid-preserving operation on a signed graph, which we call the cylinder flip, that relates certain pairs of signed-graphic representations of the same matroid.