Basis Number and Pathwidth
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909995835064320 |
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| author | Miraftab, Babak Morin, Pat Yuditsky, Yelena |
| author_facet | Miraftab, Babak Morin, Pat Yuditsky, Yelena |
| contents | We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14095 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Basis Number and Pathwidth Miraftab, Babak Morin, Pat Yuditsky, Yelena Combinatorics Discrete Mathematics We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$. |
| title | Basis Number and Pathwidth |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2601.14095 |