Optimal trees of tangles: refining the essential parts

Abstract: 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 $k \in \mathbb{N}$, 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.