The excluded 3-minors for vf-safe delta-matroids
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866910556072443904 |
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| author | Bonin, Joseph E Chun, Carolyn Noble, Steven D |
| author_facet | Bonin, Joseph E Chun, Carolyn Noble, Steven D |
| contents | Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic delta-matroids by finding the minimal obstructions, called excluded 3-minors, to membership in the class. We find the unique (up to twisted duality) excluded $3$-minor within the class of set systems for the class of vf-safe delta-matroids. In the literature, binary delta-matroids appear in many different guises, with appropriate notions of minor operations equivalent to that of $3$-minors, perhaps most notably as graphs with vertex minors. We give a direct explanation of this equivalence and show that some well-known results may be expressed in terms of $3$-minors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_01376 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The excluded 3-minors for vf-safe delta-matroids Bonin, Joseph E Chun, Carolyn Noble, Steven D Combinatorics Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic delta-matroids by finding the minimal obstructions, called excluded 3-minors, to membership in the class. We find the unique (up to twisted duality) excluded $3$-minor within the class of set systems for the class of vf-safe delta-matroids. In the literature, binary delta-matroids appear in many different guises, with appropriate notions of minor operations equivalent to that of $3$-minors, perhaps most notably as graphs with vertex minors. We give a direct explanation of this equivalence and show that some well-known results may be expressed in terms of $3$-minors. |
| title | The excluded 3-minors for vf-safe delta-matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1807.01376 |