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(h2 · log2 h)), any graph of pathwidth at most 2 (ch = O(h2)), and the yurt graph (ch = O(h4)). © 2015 Wiley Periodicals, Inc