On the Minimum Size of a Contraction-Universal Tree

TL;DR

This paper establishes a lower bound on the size of contraction-universal trees for all trees of a given size and constructs such trees close to this bound, advancing understanding of tree contraction universality.

Contribution

It proves a new lower bound on the size of contraction-universal trees and constructs near-optimal trees with fewer edges, improving previous bounds.

Findings

Lower bound of m ln(m) + (γ-1)m + O(1) edges for contraction-universal trees

Construction of contraction-universal trees with less than 1.984m edges

Advancement in understanding the minimal size of universal trees

Abstract

A tree T_uni is m-universal for the class of trees if for every tree T of size m, T can be obtained from T_uni by successive contractions of edges. We prove that a m-universal tree for the class of trees has at least mln(m) + (gamma-1)m + O(1) edges where is the Euler's constant and we build such a tree with less than mc edges for a fixed constant c = 1.984...