Refining Tree‐Decompositions so That They Display the k ‐Blocks

ABSTRACT Carmesin and Gollin proved that every finite graph has a canonical tree‐decomposition( T , V )of adhesion less thankthat efficiently distinguishes every two distinctk‐profiles, and which has the further property that every separablek‐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 thank. 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.