Nullity and Loop Complementation for Delta-Matroids

TL;DR

This paper explores the relationship between nullity, loop complementation, and delta-matroids, generalizing nullity concepts from graphs to broader set systems and characterizing delta-matroids through minimal set properties.

Contribution

It introduces a nullity measure for delta-matroids linked to symmetric difference distance and characterizes delta-matroids via equicardinality of minimal sets, extending classical graph nullity concepts.

Findings

Nullity corresponds to symmetric difference distance in delta-matroids.

Characterization of delta-matroids through equicardinality of minimal sets.

Loop complementation and pivot operations relate to null space properties.

Abstract

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension.