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 $f(H)\geq |V(H)|-1$ 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 $f(H)\leq \max\{\tfrac32|V(H)|-\tfrac92,|V(H)|-1\}$ for every wheel $H$.