Later Divergence Time graphs

Reconciliations of trees, that is, mappings μ: V(T) → V(S) ∪ E(S) from a rooted tree T, the gene tree, into another tree S, the species tree, with a given correspondence between the leaves describes e.g. the evolution of gene families or the co-evolution of hosts and parasites. Here we consider both trees “timed”, i.e., vertices u of T and x of S is associated with a timestamp τ_T(u) and τ_S(x) that respect the ancestor order of the trees and satisfy τ_T(v) = τ_S(μ(v)). Horizontal transfer corresponds to edges xy ∈ E(T) such that μ(x) and μ(y) are uncomparable w.r.t. the ancestor order in S. The identification of horizontal transfer events is a difficult problem in practical data analysis. An approach used in practise is to consider pairs of leaves (u, v) in T such that the last common ancestor of the species σ(u) and σ(v) is older than the last common ancestor of u and v. This defines a (colored, undirected) graph, the Later Divergence Time (LDT) graph, which can be estimated from data.

The presentation will be concerned with the characterization of the class of LDT graph and ask how much information on S, T, μ can be retrieved from a given LDT. We shall see that LDT graphs are properly colored cographs and determine certain “informative” sets of rooted triples that are displayed and T and S, respectively. The LDT graph, furthermore, is a subgraph of the Fitch graph, that is the graph whose edges are the pair of genes that are separated by a horizontal transfer event, in general however, it does not identify all horizontal transfer events.

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