Linear bounds on treewidth in terms of excluded planar minors
arxiv(2024)
摘要
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 Ω(klog k)
which does not contain k vertex-disjoint cycles, from which it follows that
f(H) = Ω(klog k). In particular, if f(H) is linear in
|V(H)| for graphs H from a subclass of planar graphs, it is
necessary that n-vertex graphs from the class contain at most
O(n/log(n)) 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) ≤ 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)/log(n)). 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.
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