A family of subsets of a ground set closed under the operation of taking symmetric differences is the family of cycles of a binary matroid. Its circuits are the minimal members of this collection. We use this basic property to derive binary matroids from binary matroids. In particular, we derive two
Splitting in a binary matroid
โ Scribed by T.T. Raghunathan; M.M. Shikare; B.N. Waphare
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 210 KB
- Volume
- 184
- Category
- Article
- ISSN
- 0012-365X
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โฆ Synopsis
We introduce a splitting operation for binary matroids which is a natural generalization of the splitting operation for graphs and investigate some of its basic properties. Eulerian binary matroids are characterized in terms of the splitting operation.
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In this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently,
It is shown that each binary matroid contains an odd number of maximal cycles and, as a result of this, that each element of an Eulerian binary matroid is contained in an odd number of circuits. Let M be a binary matroid with circuits ~(M) and cycles .~(M), and let ~e(M) be the set of circuits conta