Evolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree $T$ to vertices and edges of a species tree $S$. The relative timing of the last common ancestors of two extant genes (leaves of $T$) and the last common ancestors of the two species (leaves of $S$) in which they reside is indicative of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene pairs, on the other hand, require that their last common ancestors coincides with a corresponding speciation event. The relative timing information of gene and species divergences is captured by three colored graphs that have the extant genes as vertices and the species in which the genes are found as vertex colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT) graph and the prior-divergence-time (PDT) graph, which together form an edge partition of the complete graph. Here we give a complete characterization in terms of informative and forbidden triples that can be read off the three graphs and provide a polynomial time algorithm for constructing an evolutionary scenario that explains the graphs, provided such a scenario exists. While both LDT and PDT graphs are cographs, this is not true for the EDT graph in general. We show that every EDT graph is perfect. While the information about LDT and PDT graphs is necessary to recognize EDT graphs in polynomial-time for general scenarios, this extra information can be dropped in the HGT-free case. However, recognition of EDT graphs without knowledge of putative LDT and PDT graphs is NP-complete for general scenarios. We finally connect the EDT graph to the alternative definitions of orthology that have been proposed for scenarios with horizontal gene transfer. With one exception, the corresponding graphs are shown to be colored cographs.

翻译：描述一个基因家族在多个物种集合中进化过程的进化场景，包含将基因树 $T$ 的顶点映射到物种树 $S$ 的顶点和边上。两个现存基因（$T$ 的叶节点）的最近共同祖先与其所在物种（$S$ 的叶节点）的最近共同祖先之间的相对时间关系，可指示水平基因转移（HGT）和古老复制事件。而直系同源基因对则要求其最近共同祖先与对应的物种分化事件同时发生。基因与物种分歧的相对时序信息由三种着色图捕获，这些图以现存基因作为顶点，并以基因所在物种作为顶点颜色：等分歧时间图、后分歧时间图和先分歧时间图，三者共同构成完全图的一个边划分。本文给出了基于这三张图中可读取的信息性三元组和禁止性三元组的完整特征刻画，并提供了一种多项式时间算法，用于构建能够解释这些图的进化场景（若此类场景存在）。尽管后分歧时间图和先分歧时间图均为余图，但等分歧时间图通常不具备该性质。我们证明每个等分歧时间图都是完美图。在一般场景中，后分歧时间图和先分歧时间图的信息对于多项式时间内识别等分歧时间图是必要的；但在无水平基因转移的情况下，可忽略此额外信息。然而，对于一般场景，若缺乏假定后分歧时间图和先分歧时间图的知识，则识别等分歧时间图是NP完全的。最后，我们将等分歧时间图与为含水平基因转移场景提出的其他直系同源关系定义建立关联。除一种情况外，相应图均被证明为着色余图。