Several implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

几种推断水平基因转移 (HGT) 的隐式方法侧重于仅在基因所在的两个物种发生分歧后才发生分歧的基因对。这种情况定义了图的边集，即后期发散时间 (LDT) 图，其顶点对应于按物种着色的基因。我们在宽松场景的设置中研究这些图，即包含文献中所有常用的复制-转移-损失场景变体的进化场景。我们将 LDT 图表征为适当顶点着色的 cograph 的子类，并提供多项式时间识别算法以及构建解释给定 LDT 的宽松场景的算法。LDT 图中的一条边意味着两个相应的基因被至少一个 HGT 事件分开。然而，反过来却是不正确的。我们表明，完整的异种关系由 rs-Fitch 图描述，即满足顶点着色约束的完整多部分图。这类顶点着色图在多项式时间内也是可识别的。我们最终在模拟具有广泛重复、丢失和 HGT 事件的进化场景的帮助下解决了“LDT 图中包含多少关于所有 HGT 事件的信息”的问题。特别是，我们展示了一个简单的贪婪图编辑方案可用于有效检测隐含在 LDT 图中的 HGT 事件。这类顶点着色图在多项式时间内也是可识别的。我们最终在模拟具有广泛重复、丢失和 HGT 事件的进化场景的帮助下解决了“LDT 图中包含多少关于所有 HGT 事件的信息”的问题。特别是，我们展示了一个简单的贪婪图编辑方案可用于有效检测隐含在 LDT 图中的 HGT 事件。这类顶点着色图在多项式时间内也是可识别的。我们最终在模拟具有广泛重复、丢失和 HGT 事件的进化场景的帮助下解决了“LDT 图中包含多少关于所有 HGT 事件的信息”的问题。特别是，我们展示了一个简单的贪婪图编辑方案可用于有效检测隐含在 LDT 图中的 HGT 事件。