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 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(r² 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.