Refining tree-decompositions so that they display the k-blocks

Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition (T, V) of adhesion less than k that efficiently distinguishes every two distinct k-profiles, and which has the further property that every separable k-block is equal to the unique part of (T, V) in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than k. For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.