20.500.12556/RUP-22371 Linear bounds on treewidth in terms of excluded planar minors One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for $f(H)$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth $\Omega(k\log k)$ which does not contain $k$ vertex-disjoint cycles, from which it follows that $f(H) = \Omega(k\log k)$. In particular, if $f(H)$ is linear in $\lvert V(H) \rvert$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most $\lvert V(H) \rvert$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to $f(H)$≤$n$+O(√$n$) when $r$=2. In particular this bound is linear when $r$=O(√$n$/logn). We present a linear bound for $f(H)$ when $H$ is a subdivision of an $r$-edge planar graph for any constant~$r$. We also improve the best known bounds for $f(H)$ when $H$ is the wheel graph or the 4×4 grid, obtaining a bound of 160 for the latter. graph minor treewidth cycle packing grafovski minor drevesna širina pakiranje ciklov true false true Angleški jezik Slovenski jezik Članek v reviji 2026-01-05 14:56:30 2026-01-05 14:56:32 2026-01-06 03:04:00 0000-00-00 00:00:00 2025 0 0 str. 1-23 iss. 4, article no. P4.68 Vol. 32 2025 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2025-12-12 519.17 1077-8926 10.37236/12834 263444995 RAZ_Gollin_J._Pascal_2025.pdf RAZ_Gollin_J._Pascal_2025.pdf 1 628F2E4245CA6B2FAD96A7E801EAF52C 833ade8b23aac88efa0b019f8909a659816e27fde4f21d048669531e63b9fe30 dcef79fe-ea3e-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32509 https://www.combinatorics.org/ojs/index.php/eljc/article/view/v32i4p68 1 b28bbffb-ea3e-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=32508 Fakulteta za matematiko, naravoslovje in informacijske tehnologije 0 0 0