Fitch graphs $G=(X,E)$ are digraphs that are explained by $\{\emptyset, 1\}$-edge-labeled rooted trees $T$ with leaf set $X$: there is an arc $(x,y) \in E$ if and only if the unique path in $T$ that connects the last common ancestor $\mathrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label "1". In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\mathrm{lca}(x,y)$ to $y$. In this contribution, we generalize the concept of Fitch graphs and consider trees $T$ that are equipped with edge-labeling $\lambda: E\to \mathcal{P}(M)$ that assigns to each edge a subset $M'\subseteq M$ of colors. Given such a tree, we can derive a map $\varepsilon_{(T,\lambda)}$ (or equivalently a set of not necessarily disjoint binary relations), such that $i\in \varepsilon_{(T,\lambda)}(x,y)$ (or equivalently $(x,y)\in R_i$) with $x,y\in X$, if and only if there is at least one edge with color $i$ from $\mathrm{lca}(x,y)$ to $y$. The central question considered here: Is a given map $\varepsilon$ a Fitch map, i.e., is there there an edge-labeled tree $(T,\lambda)$ with $\varepsilon_{(T,\lambda)} = \varepsilon$, and thus explains $\varepsilon$? Here, we provide a characterization of Fitch maps in terms of certain neighborhoods and forbidden submaps. Further restrictions of Fitch maps are considered. Moreover, we show that the least-resolved tree explaining a Fitch map is unique (up to isomorphism). In addition, we provide a polynomial-time algorithm to decide whether $\varepsilon$ is a Fitch map and, in the affirmative case, to construct the (up to isomorphism) unique least-resolved tree $(T^*,\lambda^*)$ that explains $\varepsilon$.

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广义惠誉图 II：由边标记树解释的二元关系集

Fitch 图 $G=(X,E)$ 是由 $\{\emptyset, 1\}$-edge-labeled root tree $T$ 和叶子集 $X$ 解释的有向图：存在一条弧 $(x ,y) \in E$ 当且仅当连接 $x$ 和 $y$ 的最后一个共同祖先 $\mathrm{lca}(x,y)$ 的 $T$ 中的唯一路径包含在至少有一个标签为“1”的边。在实践中，Fitch 图表示异种关系，即基因对 $x$ 和 $y$，其中沿从 $\mathrm{lca}(x,y)$ 到 $y$ 的路径发生水平基因转移。在这个贡献中，我们概括了 Fitch 图的概念，并考虑了配备边标记 $\lambda 的树 $T$：E\to \mathcal{P}(M)$ 为每个边分配一个子集 $M' \subseteq M$ 的颜色。给定这样一棵树，我们可以导出一个映射 $\varepsilon_{(T, \lambda)}$（或等效的一组不一定不相交的二元关系），使得 $i\in \varepsilon_{(T,\lambda)}(x,y)$（或等效的 $(x,y)\在 R_i$) 和 $x,y\in X$ 中，当且仅当从 $\mathrm{lca}(x,y)$ 到 $y$ 至少有一条带颜色 $i$ 的边。这里考虑的中心问题是：给定的地图 $\varepsilon$ 是否是 Fitch 地图，即是否存在带有 $\varepsilon_{(T,\lambda)} = \ 的边标记树 $(T,\lambda)$ varepsilon$，从而解释了 $\varepsilon$? 在这里，我们根据某些邻域和禁止子图提供了 Fitch 地图的特征。考虑了惠誉地图的进一步限制。此外，我们证明了解释 Fitch 映射的最低分辨率树是唯一的（直到同构）。此外，我们提供了一个多项式时间算法来确定 $\varepsilon$ 是否是 Fitch 映射，并且，