Lassoing and corraling rooted phylogenetic trees

TL;DR

This paper investigates the conditions under which a rooted phylogenetic tree can be uniquely reconstructed from partial distance data, providing characterizations for different notions of uniqueness relevant to genome analysis.

Contribution

It introduces four formal types of lassos for rooted phylogenetic trees and characterizes them using child-edge graphs, advancing understanding of tree reconstruction from partial data.

Findings

All four lasso types coincide for binary trees.

Characterizations are given in terms of child-edge graphs.

The results inform when partial distances suffice for unique tree reconstruction.

Abstract

The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However even with modern sequencing technologies the distances on the individuals required for the construction of such a structure may not always be reliable making it tempting to exclude them from an analysis. This, in turn, results in an input set for dendogram construction that consists of only partial distance information which raises the following fundamental question. For what subset of its leaf set can we reconstruct uniquely the dendogram from the distances that it induces on that subset. By formalizing a dendogram in terms of an edge-weighted, rooted phylogenetic tree on a pre-given finite set X with |X|>2 whose edge-weighting is equidistant and a set of partial distances on X in terms of a set L of 2-subsets of X, we investigate this problem in terms of when such a tree is lassoed, that is, uniquely determined by the elements in L. For this we consider four different formalizations of the idea of "uniquely determining" giving rise to four distinct types of lassos. We present characterizations for all of them in terms of the child-edge graphs of the interior vertices of such a tree. Our characterizations imply in particular that in case the tree in question is binary then all four types of lasso must coincide.