The excluded minors for three classes of 2-polymatroids having special types of natural matroids
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914901382922240 |
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| author | Bonin, Joseph E. Long, Kevin |
| author_facet | Bonin, Joseph E. Long, Kevin |
| contents | If $\mathcal{C}$ is a minor-closed class of matroids, the class $\mathcal{C}'$ of integer polymatroids whose natural matroids are in $\mathcal{C}$ is also minor closed, as is the class $\mathcal{C}'_k$ of $k$-polymatroids in $\mathcal{C}'$. We find the excluded minors for $\mathcal{C}'_2$ when $\mathcal{C}$ is (i) the class of binary matroids, (ii) the class of matroids with no $M(K_4)$-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class $\mathcal{C}$ has finitely many excluded minors, but that is true of $\mathcal{C}'_2$ only in case (ii). We also introduce the $k$-natural matroid, a variant of the natural matroid for a $k$-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_03797 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The excluded minors for three classes of 2-polymatroids having special types of natural matroids Bonin, Joseph E. Long, Kevin Combinatorics If $\mathcal{C}$ is a minor-closed class of matroids, the class $\mathcal{C}'$ of integer polymatroids whose natural matroids are in $\mathcal{C}$ is also minor closed, as is the class $\mathcal{C}'_k$ of $k$-polymatroids in $\mathcal{C}'$. We find the excluded minors for $\mathcal{C}'_2$ when $\mathcal{C}$ is (i) the class of binary matroids, (ii) the class of matroids with no $M(K_4)$-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class $\mathcal{C}$ has finitely many excluded minors, but that is true of $\mathcal{C}'_2$ only in case (ii). We also introduce the $k$-natural matroid, a variant of the natural matroid for a $k$-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality. |
| title | The excluded minors for three classes of 2-polymatroids having special types of natural matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2209.03797 |