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arXiv · 1010.4497

Nullity and Loop Complementation for Delta-Matroids

Robert Brijder·Hendrik Jan Hoogeboom
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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.

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Robert Brijder, Hendrik Jan Hoogeboom. 2010-10-21. Nullity and Loop Complementation for Delta-Matroids. https://doi.org/10.1137/110854692

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