Nullity and Loop Complementation for Delta-Matroids

Details Nullity and Loop Complementation for Delta-Matroids Nullity and Loop Complementation for Delta-Matroids

2010-10-21

arxiv.org/abs/1010.4497v4

Mathematics

Combinatorics

Changes w.r.t. v3: Section 8 is revised and in addition a few small changes are made in the rest of the paper. 17 pages, no fi

Scientific paper

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 w.r.t. 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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