Forbidden minors for bounded treewidth graphs This question is similar to one of my previous questions. It is known that $K_{t+2}$ is a forbidden minor for graphs of treewidth at most $t$. 


  Is there a nicely-constructed, parameterized, infinite family of graphs (other than complete graphs and grid graphs) that are minimal forbidden minors for graphs of every treewidth. In other words, is there an explicit graph $G_r$ on $r$ vertices (which is not a complete graph) such that $G_r$ is a forbidden minor for graphs of treewidth at most $r$, where $r$ is a function of $t$ ?


The complete sets of forbidden minors are known for graphs of treewidth at most three. See this Wikipedia article for more details.


  Is the complete set of forbidden minors of graphs of treewidth at most four known ?
In Sparse obstructions and exact treewidth determination, Lucena states that in the PhD thesis of Sanders, "75 or so minimal forbidden minors for treewidth $\leq 4$ are given, and it is believed though not proven that this may constitute the entire obstruction set."