Treedepth and 2-treedepth in graphs with no long induced paths
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909785487572992 |
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| author | Hodor, Jędrzej Illingworth, Freddie Mazur, Tomasz |
| author_facet | Hodor, Jędrzej Illingworth, Freddie Mazur, Tomasz |
| contents | Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every graph, 2-treedepth is at most the treedepth but can be much smaller: long paths have arbitrary treedepth but 2-treedepth equal to 2. We prove a converse showing that every graph with no induced path on $t$ vertices and 2-treedepth at most $k$ has treedepth at most $g(k, t)$. In fact, we determine the value of the function $g$ up to a multiplicative factor of 2.
Additionally, we give asymptotically tight bounds for the problem of forcing long induced paths in graphs with long paths and bounded 2-treedepth or bounded pathwidth. The latter result answers a question of Hilaire and Raymond (E-JC, 2024). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Treedepth and 2-treedepth in graphs with no long induced paths Hodor, Jędrzej Illingworth, Freddie Mazur, Tomasz Combinatorics Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every graph, 2-treedepth is at most the treedepth but can be much smaller: long paths have arbitrary treedepth but 2-treedepth equal to 2. We prove a converse showing that every graph with no induced path on $t$ vertices and 2-treedepth at most $k$ has treedepth at most $g(k, t)$. In fact, we determine the value of the function $g$ up to a multiplicative factor of 2. Additionally, we give asymptotically tight bounds for the problem of forcing long induced paths in graphs with long paths and bounded 2-treedepth or bounded pathwidth. The latter result answers a question of Hilaire and Raymond (E-JC, 2024). |
| title | Treedepth and 2-treedepth in graphs with no long induced paths |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.04445 |