Simultaneous separation in bounded degree trees

2026-02-26

AI Summary

This study addresses the synchronized single-edge cut problem across multiple bounded-degree trees sharing a common vertex set labeled by taxa, aiming to maximize the size of the smaller labeled subset induced in each tree after the cut. By integrating combinatorial methods, tree partitioning theory, and asymptotic analysis, the work establishes the exact asymptotic optimum $f(r,2) = 1/(2r)$ for two trees and $f(3,3) = 2/27$ for three cubic trees. These results imply that for any pair (or triple) of binary phylogenetic trees on $n$ taxa, there always exists a synchronized cut yielding parts each containing at least $n/6$ (or $2n/27$) taxa, with these bounds being tight. This resolves a fundamental theoretical question concerning tree dissimilarity in phylogenetics.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

It follows from a classical result of Jordan that every tree with maximum degree at most $r$ containing a vertex set labeled by $[n]$, has a single-edge cut which separates two subsets $A,B \subset [n]$ for which $\min\{|A|,|B|\} \ge (n-1)/r$. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of {\em several} trees: Given $k$ trees with maximum degree at most $r$, containing a common vertex set labeled by $[n]$, we ask for a single-edge cut in each tree which maximizes $min\{|A|,|B|\}$ where $A,B \subset [n]$ are separated by the corresponding cut at each tree. Denoting this maximum by $f(r,k,n)$ and considering the limit $f(r,k) = \lim_{n \rightarrow \infty} f(r,k,n)/n$ (which is shown to always exist) we determine that $f(r,2)=\frac{1}{2r}$ and determine that $f(3,3)=\frac{2}{27}$, which is already quite intricate. The case $r=3$ is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over $n$ taxa have a split at each tree which separates two taxa sets of order at least $n/6$ (resp. $2n/27$), and these bounds are asymptotically tight.

Problem

• tree separation
• bounded degree trees
• simultaneous cut
• phylogenetic trees
• combinatorial optimization

Innovation

• simultaneous tree separation
• bounded degree trees
• phylogenetic trees
• combinatorial optimization
• asymptotic tight bound