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

Abstract

Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition $(T, \mathcal{V})$ 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.

Figures (1)

• Figure 1: A component $C \in \mathcal{C}'$ and the arising separation $(X_C,Y_C)$. The separations $(A,B) \in \sigma_C$ are indicated with solid lines, the separations $(A', B') \in \sigma \smallsetminus \sigma_C$ are indicated with dashed lines. The component $\tilde{C}_A$ of $G - (A \cap B)$ is contained in $C$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• proof : Proof of \ref{['thm:DisplayingBlocksRefining']}