Combinatorial properties of triplet covers for binary trees

Grunewald, Stefan, Huber, Katharina T., Moulton, Vincent and Steel, Mike (2018) Combinatorial properties of triplet covers for binary trees. Advances in Applied Mathematics, 99. pp. 59-82. ISSN 0196-8858

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    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'.

    Item Type: Article
    Uncontrolled Keywords: phylogenetic tree,triplet cover, tree-distances, hall's theorem,ample patchwork, shellability
    Faculty \ School: Faculty of Science > School of Computing Sciences
    ?? RGCB ??
    University of East Anglia > Faculty of Science > Research Groups > Computational Biology (subgroups are shown below) > Phylogenetics
    University of East Anglia > Faculty of Science > Research Groups > Computational Biology (subgroups are shown below) > Computational biology of RNA
    Depositing User: Pure Connector
    Date Deposited: 16 Apr 2018 14:30
    Last Modified: 17 May 2018 15:32
    URI: https://ueaeprints.uea.ac.uk/id/eprint/66773
    DOI:

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