Combinatorial properties of triplet covers for binary trees

TL;DR

This paper proves that triplet covers uniquely determine binary trees with edge lengths from leaf-to-leaf distances and explores their combinatorial properties such as minimality, sparsity, and shellability.

Contribution

It answers an open question by showing triplet covers uniquely determine the tree and its edge lengths, and analyzes their combinatorial structures.

Findings

Triplet covers uniquely determine the tree and edge lengths.

Characterization of minimal, sparse, and shellable triplet covers.

Structural insights into the properties of triplet covers.

Abstract

It is a classical result that an unrooted tree $T$ having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of $T$ has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in $T$ is the following: for each non-leaf vertex $v$, choose a leaf in each of the three components of $T-v$, group these three leaves into three pairs, and take the union of this set over all choices of $v$. This forms a so-called 'triplet cover' for $T$. In the first part of this paper we answer an open question (from 2012) by showing that the induced leaf-to-leaf distances for any triplet cover for $T$ uniquely determine $T$ and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, 'sparse', and 'shellable'.