Tree-width of a graph excluding an apex-forest or a wheel as a minor

The Grid Minor Theorem states that for every planar graph H, there exists a smallest integer f (H) such that every graph with tree-width at least f (H) contains H as a minor. The only known lower bounds on f (H) beyond the trivial bound f (H) |V (H) |-1 come from the maximum number of disjoint cycles in H. In this paper, we study f (H) for planar graphs H with no two disjoint cycles. We prove that f (H) =|V (H) |-1 for every apex-forest H. This result improves a bound of Leaf and Seymour and contains all known large graphs H meeting the trivial lower bound to our knowledge. We also prove that f (H) \32|V (H) |-92, |V (H) |-1\ for every wheel H.