Optimal trees of tangles: refining the essential parts

We combine the two fundamental fixed-order tangle theorems of Robertson and Seymour into a single theorem that implies both, in a best possible way. We show that, for every k ∈ ℕ , every tree-decomposition of a graph G which efficiently distinguishes all its k -tangles can be refined to a tree-decomposition whose parts are either too small to be home to a k -tangle, or as small as possible while being home to a k -tangle.