Graph parameters that are coarsely equivalent to tree-length
Feodor F. Dragan
TL;DR
The paper investigates when a graph has small tree-length $tl(G)$ by showing that a broad family of graph parameters are coarsely equivalent to $tl(G)$, often via layered partitions and their cluster metrics. It provides simpler proofs and stronger constants for known equivalences, and extends the list of coarsely equivalent parameters to include bottleneck constants, McCarty-width, brambles/Helly families, distance-$k$-approximating trees, $K_3$-minor fatness, and cycle-bridging properties, among others. It also strengthens the connections between tree-length and tree-embedding notions, including quasi-isometries to trees and additive/multiplicative distortion bounds, and yields algorithmic implications such as balanced disk separators and linear-time constructions. Overall, the work unifies diverse graph-analytic frameworks under the lens of tree-length, enabling transfer of results and facilitating improved bounds and embeddings with and without Steiner points. The results have significance for designing approximation algorithms and understanding the structural geometry of graphs through a common, coarse-tuning parameter.
Abstract
Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length of $G$ is the minimum of the length, over all tree-decompositions of $G$. We present simpler and sometimes with better bounds proofs for those known in literature results and further extend this list of graph parameters coarsely equivalent to tree-length. Among other new results, we show that the tree-length of a graph $G$ is small if and only if for every bramble ${\cal F}$ (or every Helly family of connected subgraphs ${\cal F}$, or every Helly family of paths ${\cal F}$) of $G$, there is a disk in $G$ with small radius that intercepts all members of ${\cal F}$. Furthermore, the tree-length of a graph $G$ is small if and only if $G$ can be embedded with a small additive distortion to an unweighted tree with the same vertex set as in $G$ (not involving any Steiner points). Additionally, we introduce a new natural "bridging`` property for cycles, which generalizes a known property of cycles in chordal graphs, and show that it also coarsely defines the tree-length.