Extensions and Deletions of matroid classes closed under flats
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913171114033152 |
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| author | Singh, Jagdeep Sivaraman, Vaidy |
| author_facet | Singh, Jagdeep Sivaraman, Vaidy |
| contents | We call a class of matroids hereditary if it is closed under restriction to flats. For a hereditary class $\mathcal{M}$, its extension class consists of all matroids in $\mathcal{M}$ together with their single-element extensions. The deletion class consists of all matroids in $\mathcal{M}$ along with their single-element deletions.
We prove that if $\mathcal{M}$ has finitely many forbidden flats, then the forbidden flats for its extension class have bounded rank. For $GF(q)$-representable matroids where $q$ is in $\{2,3\}$, we exploit correspondence with $2$-colorings of projective geometries to establish the analogous result for the deletion class. We also note the consequences for hereditary classes of graphs, discussing the interplay of graphs and matroids. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_15496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extensions and Deletions of matroid classes closed under flats Singh, Jagdeep Sivaraman, Vaidy Combinatorics 05C75, 05B35 We call a class of matroids hereditary if it is closed under restriction to flats. For a hereditary class $\mathcal{M}$, its extension class consists of all matroids in $\mathcal{M}$ together with their single-element extensions. The deletion class consists of all matroids in $\mathcal{M}$ along with their single-element deletions. We prove that if $\mathcal{M}$ has finitely many forbidden flats, then the forbidden flats for its extension class have bounded rank. For $GF(q)$-representable matroids where $q$ is in $\{2,3\}$, we exploit correspondence with $2$-colorings of projective geometries to establish the analogous result for the deletion class. We also note the consequences for hereditary classes of graphs, discussing the interplay of graphs and matroids. |
| title | Extensions and Deletions of matroid classes closed under flats |
| topic | Combinatorics 05C75, 05B35 |
| url | https://arxiv.org/abs/2403.15496 |