The space of equidistant phylogenetic cactuses

Katharina T. Huber, Vincent Moulton, Megan Owen, Andreas Spillner, Katherine St. John

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An equidistant X-cactus is a type of rooted, arc-weighted, directed acyclic graph with leaf set X, that is used in biology to represent the evolutionary history of a set X of species. In this paper, we introduce and investigate the space of equidistant X-cactuses. This space contains, as a subset, the space of ultrametric trees on X that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning ranked rooted X-cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of X that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted X-trees in terms of partitions of X, which provides an alternative proof that the space of ultrametric trees on X is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant X-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.

Original languageEnglish
Pages (from-to)1-32
Number of pages32
JournalAnnals of Combinatorics
Early online date9 Jun 2023
Publication statusPublished - Mar 2024


  • phylogenetic network
  • network space
  • combinatorial encoding
  • CAT(0)-metric space
  • Phylogenetic network
  • Combinatorial encoding
  • Network space
  • 92D15
  • 05C90
  • 06A06
  • 52B70

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