Tree covers of size $2$ for the Euclidean plane

TL;DR

This paper proves that any finite set of points in the Euclidean plane can be covered by just two trees with a constant stretch factor, advancing understanding of tree covers in metric spaces.

Contribution

It establishes the existence of a 2-tree cover with constant stretch for Euclidean plane point sets, and provides lower bounds for higher dimensions.

Findings

Existence of 2-tree cover with constant stretch in the Euclidean plane.

Lower bounds on the number of trees needed in higher dimensions.

Extension of tree cover concepts to Euclidean metrics.

Abstract

For a given metric space $(P,\phi)$, a tree cover of stretch $t$ is a collection of trees on $P$ such that edges $(x,y)$ of trees receive length $\phi(x,y)$, and such that for any pair of points $u,v\in P$ there is a tree $T$ in the collection such that the induced graph distance in $T$ between $u$ and $v$ is at most $t\phi(u,v).$ In this paper, we show that, for any set of points $P$ on the Euclidean plane, there is a tree cover consisting of two trees and with stretch $O(1).$ Although the problem in higher dimensions remains elusive, we manage to prove that for a slightly stronger variant of a tree cover problem we must have at least $(d+1)/2$ trees in any constant stretch tree cover in $\mathbb R^d$.