Tree-width of a graph excluding an apex-forest or a wheel as a minor
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| Format: | Preprint |
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2025
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| _version_ | 1866909783494230016 |
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| author | Liu, Chun-Hung Yoo, Youngho |
| author_facet | Liu, Chun-Hung Yoo, Youngho |
| contents | The Grid Minor Theorem states that for every planar graph $H$, there exists a smallest integer $f(H)$ such that every graph with tree-width at least $f(H)$ contains $H$ as a minor. The only known lower bounds on $f(H)$ beyond the trivial bound $f(H)\geq |V(H)|-1$ come from the maximum number of disjoint cycles in $H$. In this paper, we study $f(H)$ for planar graphs $H$ with no two disjoint cycles. We prove that $f(H)=|V(H)|-1$ for every apex-forest $H$. This result improves a bound of Leaf and Seymour and contains all known large graphs $H$ meeting the trivial lower bound to our knowledge. We also prove that $f(H)\leq \max\{\tfrac32|V(H)|-\tfrac92,|V(H)|-1\}$ for every wheel $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09895 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tree-width of a graph excluding an apex-forest or a wheel as a minor Liu, Chun-Hung Yoo, Youngho Combinatorics The Grid Minor Theorem states that for every planar graph $H$, there exists a smallest integer $f(H)$ such that every graph with tree-width at least $f(H)$ contains $H$ as a minor. The only known lower bounds on $f(H)$ beyond the trivial bound $f(H)\geq |V(H)|-1$ come from the maximum number of disjoint cycles in $H$. In this paper, we study $f(H)$ for planar graphs $H$ with no two disjoint cycles. We prove that $f(H)=|V(H)|-1$ for every apex-forest $H$. This result improves a bound of Leaf and Seymour and contains all known large graphs $H$ meeting the trivial lower bound to our knowledge. We also prove that $f(H)\leq \max\{\tfrac32|V(H)|-\tfrac92,|V(H)|-1\}$ for every wheel $H$. |
| title | Tree-width of a graph excluding an apex-forest or a wheel as a minor |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.09895 |