The symmetric strong circuit elimination property
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909715471007744 |
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| author | Cho, Christine Oxley, James Wang, Suijie |
| author_facet | Cho, Christine Oxley, James Wang, Suijie |
| contents | If $C_1$ and $C_2$ are circuits in a matroid $M$ with $e_1$ in $C_1-C_2$ and $e$ in $C_1\cap C_2$, then $M$ has a circuit $C_3$ such that $e\in C_3\subseteq (C_1\cup C_2)-e$. This strong circuit elimination axiom is inherently asymmetric. A matroid $M$ has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and $e_2\in C_2-C_1$, there is a circuit $C_3'$ with $\{e_1,e_2\}\subseteq C_3'\subseteq (C_1\cup C_2)-e$. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The symmetric strong circuit elimination property Cho, Christine Oxley, James Wang, Suijie Combinatorics 05B35 If $C_1$ and $C_2$ are circuits in a matroid $M$ with $e_1$ in $C_1-C_2$ and $e$ in $C_1\cap C_2$, then $M$ has a circuit $C_3$ such that $e\in C_3\subseteq (C_1\cup C_2)-e$. This strong circuit elimination axiom is inherently asymmetric. A matroid $M$ has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and $e_2\in C_2-C_1$, there is a circuit $C_3'$ with $\{e_1,e_2\}\subseteq C_3'\subseteq (C_1\cup C_2)-e$. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE. |
| title | The symmetric strong circuit elimination property |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2508.00132 |