Iskanje po repozitoriju

Opis: 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.
Ključne besede: graph minor, treewidth, cycle packing
Objavljeno v RUP: 05.01.2026; Ogledov: 409; Prenosov: 2
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