Large induced subgraph with a given pathwidth in outerplanar graphs
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908383511052288 |
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| author | Matsumoto, Naoki Yashima, Takamasa |
| author_facet | Matsumoto, Naoki Yashima, Takamasa |
| contents | A long-standing conjecture by Albertson and Berman states that every planar graph of order $n$ has an induced forest with at least $\lceil \frac{n}{2} \rceil$ vertices. As a variant of this conjecture, Chappell conjectured that every planar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n}{9} \rceil$ vertices. Pelsmajer proved that every outerplanar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n+2}{7}\rceil$ vertices and this bound is sharp. In this paper, we investigate the order of induced subgraphs of outerplanar graphs with a given pathwidth. The above result by Pelsmajer implies that every outerplanar graph of order $n$ has an induced subgraph with pathwidth one and at least $\lceil \frac{4n+2}{7}\rceil$ vertices. We extend this to obtain a result on the maximum order of any outerplanar graph with at most a given pathwidth. We also give its upper bound which generalizes Pelsmajer's construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23162 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large induced subgraph with a given pathwidth in outerplanar graphs Matsumoto, Naoki Yashima, Takamasa Discrete Mathematics Combinatorics 05C10, 05C70 A long-standing conjecture by Albertson and Berman states that every planar graph of order $n$ has an induced forest with at least $\lceil \frac{n}{2} \rceil$ vertices. As a variant of this conjecture, Chappell conjectured that every planar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n}{9} \rceil$ vertices. Pelsmajer proved that every outerplanar graph of order $n$ has an induced linear forest with at least $\lceil \frac{4n+2}{7}\rceil$ vertices and this bound is sharp. In this paper, we investigate the order of induced subgraphs of outerplanar graphs with a given pathwidth. The above result by Pelsmajer implies that every outerplanar graph of order $n$ has an induced subgraph with pathwidth one and at least $\lceil \frac{4n+2}{7}\rceil$ vertices. We extend this to obtain a result on the maximum order of any outerplanar graph with at most a given pathwidth. We also give its upper bound which generalizes Pelsmajer's construction. |
| title | Large induced subgraph with a given pathwidth in outerplanar graphs |
| topic | Discrete Mathematics Combinatorics 05C10, 05C70 |
| url | https://arxiv.org/abs/2505.23162 |