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Characterizing path-length matrices of unrooted binary trees

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  • Catanzaro, Daniele

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

  • Pesenti, Raffaele
  • Ronco, Roberto

Abstract

We extend some recent results on the necessary and sufficient conditions that a symmetric integer matrix of order n ≥3 must satisfy to encode the Path-Length Matrix (PLM) of a Unrooted Binary Tree (UBT) with n leaves. This problem is at the core of the combinatorics of the Balanced Minimum Evolution Problem, a NP-hard problem much studied in the literature on molecular phylogenetics. We show that, for any natural 3 ≤n ≤11, a reduced set of known conditions, excluding Buneman’ strong four-point conditions, is both necessary and sufficient to characterize PLMs of UBTs. In addition, we present a second and more general characterization based solely on linear conditions derived from the topological properties of UBTs.

Suggested Citation

  • Catanzaro, Daniele & Pesenti, Raffaele & Ronco, Roberto, 2024. "Characterizing path-length matrices of unrooted binary trees," LIDAM Discussion Papers CORE 2024028, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2024028
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