References & Citations
Mathematics > Combinatorics
Title: Refining tree-decompositions so that they display the k-blocks
(Submitted on 28 Mar 2024)
Abstract: Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition $(T, \mathcal{V})$ 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 $(T, \mathcal{V})$ 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.
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