Low Polynomial Exclusion of Planar Graph Patterns

The celebrated grid exclusion theorem states that for every h-vertex planar graph H , there is a constant ch such that if a graph G does not contain H as a minor then G has treewidth at most ch. We are looking for patterns of H where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel (ch = O(h)), the double wheel (ch = O(h 2 · log h)), any graph of pathwidth at most 2 (ch = O(h )), and the yurt graph (ch = O(h )).