Detachable pairs in $3$-connected matroids and simple $3$-connected graphs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912583989067776 |
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| author | Brettell, Nick Semple, Charles Toft, Gerry |
| author_facet | Brettell, Nick Semple, Charles Toft, Gerry |
| contents | Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following holds: $M$ has a detachable pair, $M$ has a $3$-element circuit or cocircuit, or $M$ is a spike. We address the case where $M$ has a $3$-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple $3$-connected graph $G$ with $|E(G)| \ge 13$ has a pair of edges $\{e,f\}$ such that $G/e/f$ or $G \backslash e\backslash f$ is simple and $3$-connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Detachable pairs in $3$-connected matroids and simple $3$-connected graphs Brettell, Nick Semple, Charles Toft, Gerry Combinatorics 05B35 Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following holds: $M$ has a detachable pair, $M$ has a $3$-element circuit or cocircuit, or $M$ is a spike. We address the case where $M$ has a $3$-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple $3$-connected graph $G$ with $|E(G)| \ge 13$ has a pair of edges $\{e,f\}$ such that $G/e/f$ or $G \backslash e\backslash f$ is simple and $3$-connected. |
| title | Detachable pairs in $3$-connected matroids and simple $3$-connected graphs |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2409.13834 |