Refining Tree‐Decompositions so That They Display the <i>k</i>‐Blocks

ABSTRACT Carmesin and Gollin proved that every finite graph has a canonical tree‐decomposition of adhesion less than that efficiently distinguishes every two distinct ‐profiles, and which has the further property that every separable ‐block is equal to the unique part of 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 . 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.