2017/07/25 by Stefan Gruenewald, Katharina T. Huber, Gruenewald, Stefan +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Binary number #Binary tree #Coding theory and cryptography #Combinatorics #Combinatorics (math.CO) #Computer science #Cover (algebra) #Degree (music) #Enhanced Data Rates for GSM Evolution #FOS: Mathematics #Graph #Limits and Structures in Graph Theory #Mathematics #Physics #Tree (set theory) #Vertex (graph theory) #math.CO
paper · pdf · doi:10.48550/arxiv.1707.07908
published in arXiv (Cornell University) (Cornell University) · 32 pages, 5 figures
arxiv created 2017/07/25 · openalex publication_date 2017/07/25 · arxiv updated 2017/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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'.