Chordal matroids arising from generalized parallel connections II
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915108598317056 |
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| author | Douthitt, James Dylan Oxley, James |
| author_facet | Douthitt, James Dylan Oxley, James |
| contents | In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal matroids as those matroids that can be constructed from projective geometries over $GF(q)$ by a sequence of generalized parallel connections across projective geometries over $GF(q)$. Our main result showed that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In this paper, we show that the class of $GF(2)$-chordal matroids coincides with the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or $M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02099 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chordal matroids arising from generalized parallel connections II Douthitt, James Dylan Oxley, James Combinatorics 05B35, 05C75 In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal matroids as those matroids that can be constructed from projective geometries over $GF(q)$ by a sequence of generalized parallel connections across projective geometries over $GF(q)$. Our main result showed that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In this paper, we show that the class of $GF(2)$-chordal matroids coincides with the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or $M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices. |
| title | Chordal matroids arising from generalized parallel connections II |
| topic | Combinatorics 05B35, 05C75 |
| url | https://arxiv.org/abs/2405.02099 |