  One of the major results of [N. Robertson and P. D. Seymour. Graph minors.
XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110,
1995], also known as the weak structure theorem, revealed the local structure
of graphs excluding some graph as a minor: each such graph $G$ either has small
treewidth or contains the subdivision of a wall that can be arranged
"bidimensionally" inside $G$, given that some small set of vertices are
removed. We prove an optimized version of that theorem where (i) the relation
between the treewidth of the graph and the height of the wall is linear (thus
best possible) and (ii) the number of vertices to be removed is minimized.
