Minimum triplet covers of binary phylogenetic $X$-trees

TL;DR

This paper investigates the minimal sets of leaf pairs, called triplet covers, that uniquely determine binary phylogenetic trees and their branch lengths, providing new characterizations and properties of such minimal covers.

Contribution

It introduces a characterization of minimum triplet covers via 2-trees and proves they are shellable, ensuring unique tree reconstruction from their distances.

Findings

Minimum triplet covers correspond to 2-trees.

Minimum triplet covers are shellable.

These covers uniquely determine the tree and branch lengths.

Abstract

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any `triplet cover' of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are `shellable' and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths.