On Determining if Tree-based Networks Contain Fixed Trees

Abstract

We address an open question of Francis and Steel about phylogenetic networks and trees. They give a polynomial time algorithm to decide if a phylogenetic network, N , is tree-based and pose the problem: given a fixed tree T and network N , is N based on T ? We show that it is N P -hard to decide, by reduction from 3-Dimensional Matching (3DM), and further, that the problem is fixed parameter tractable.

Ranking top-k trees in tree-based phylogenetic networks

“…Let e i be the unit vector such that i -th component is one and the others are all zeros. Also, for each τ ∈ Ω \ {τ (1) }, let id(τ) be the first index such that the i -th component of v is strictly greater than one and let e(τ) := e id(τ) . For example, τ = (1 1 1 5 8) gives e(τ) = (0 0 0 1 0).…”

“…Lemma 5.3. Let (Ω, ≤ * ) be the support tree ranking for a tree-based phylogenetic X -network N with associated probability w : A(N ) → (0, 1] and let Γ be a graph with V (Γ) = Ω and A(Γ) = {(τ, τ ) ∈ Ω × (Ω \ {τ (1)…”

“…Then, Γ is a spanning tree of the Hasse diagram of (Ω, ≤) such that τ (1) is the root of Γ and (τ, τ ) ∈ A(Γ) implies τ ≤ * τ .…”

“…While Francis and Steel's work was followed by many studies (e.g., [1,3,4,5,7,11,13]), Hayamizu's recent work [8] significantly advanced our understanding of how tree-based networks could be useful in contemporary phylogenetic analysis. In fact, Hayamizu's structure theorem has derived a series of linear-time and linear-delay algorithms for many basic problems (e.g., counting, enumeration and optimisation) on support trees, and has thus enabled various data analysis using tree-based phylogenetic networks (see [8] for details).…”

New characterisations of tree-based networks and proximity measures

“…Intuitively, a phylogenetic network is tree-based if it can be obtained from a phylogenetic tree T by simply adding edges whose end-vertices subdivide edges of T . Formalised and studied in [5], tree-based networks have since been studied in a number of recent papers [1,7,9,11,12], in a variety of contexts.…”