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

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.