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The Family of Bicircular Matroids Closed Under Duality

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Abstract

We characterize the 3-connected members of the intersection of the class of bicircular and cobicircular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors.

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Notes

  1. This check can be done by hand or by using the SageMath software package. One way to represent the bicircular matroid of a graph G in SageMath is as follows. Define a \(\mathbb Q\)-matrix A whose rows are indexed by V(G) and whose columns are indexed by E(G). The column corresponding to a link e having endpoints in rows i and j should have a \(-1\) in row i, a prime number \(p_e\) unique to e in row j, and zeros in all other rows. The column corresponding to a loop incident to vertex v should be the elementary column vector corresponding to the row for v. Now \(M(A)=B(G)\).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

    Vaidy Sivaraman

  2. Department of Mathematics and Statistics, Wright State University, Dayton, OH, 45435, USA

    Daniel Slilaty

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  1. Vaidy Sivaraman
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  2. Daniel Slilaty
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Sivaraman, V., Slilaty, D. The Family of Bicircular Matroids Closed Under Duality. Graphs and Combinatorics 38, 24 (2022). https://doi.org/10.1007/s00373-021-02413-7

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