Contraction obstructions for treewidth

We provide two parameterized graphs Γ k , Π k with the following property: for every positive integer k, there is a constant c k such that every graph G with treewidth at least c k , contains one of K k , Γ k , Π k as a contraction, where K k is a complete graph on k vertices. These three parameterized graphs can be seen as “obstruction patterns” for the treewidth with respect to the contraction partial ordering. We also present some refinements of this result along with their algorithmic consequences.