Spanning and Metric Tree Covers Parameterized by Treewidth
Michael Elkin, Idan Shabat

TL;DR
This paper introduces a new framework for spanning and metric tree covers parameterized by treewidth and separator size, achieving a smooth tradeoff between stretch and the number of trees, with applications to spanners and routing.
Contribution
It presents a unified approach to construct tree covers with adjustable stretch and size based on graph parameters like treewidth and separators, improving previous bounds.
Findings
Achieves tree covers with stretch O(k) and size depending on s(n) or t(n).
Provides spanning tree covers with stretch O(k log log n).
Improves path-reporting spanners, emulators, and routing schemes.
Abstract
Given a graph , a tree cover is a collection of trees , such that for every pair of vertices there is a tree that contains a path with a small stretch. If the trees are sub-graphs of , the tree cover is called a spanning tree cover. If these trees are HSTs, it is called an HST cover. In a seminal work, Mendel and Naor [2006] showed that for any parameter , there exists an HST cover, and a non-spanning tree cover, with stretch and with trees. Abraham et al. [2020] devised a spanning version of this result, albeit with stretch . For graphs of small treewidth , Gupta et al. [2004] devised an exact spanning tree cover with trees, and Chang et al. [2-23] devised a -approximate non-spanning tree cover with…
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Stochastic processes and statistical mechanics
