A network $N$ on a finite set $X$, $|X|\geq 2$, is a connected directed acyclic graph with leaf set $X$ in which every root in $N$ has outdegree at least 2 and no vertex in $N$ has indegree and outdegree equal to 1; $N$ is arboreal if the underlying unrooted, undirected graph of $N$ is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set $X$ of species whose ancestors have exchanged genes in the past. For $M$ some arbitrary set of symbols, $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if there exists some arboreal network $N$ whose vertices with outdegree two or more are labelled by elements in $M$ and so that $d(\{x,y\})$, $\{x,y\} \in {X \choose 2}$, is equal to the label of the least common ancestor of $x$ and $y$ in $N$ if this exists and $\odot$ else. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics and cograph theory. In this paper we show that a map $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if and only if $d$ satisfies certain 3- and 4-point conditions and the graph with vertex set $X$ and edge set consisting of those pairs $\{x,y\} \in {X \choose 2}$ with $d(\{x,y\}) \neq \odot$ is Ptolemaic. To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network $N$ on $X$, where this is the graph with vertex set $X$ and edge set consisting of those $\{x,y\} \in {X \choose 2}$ such that $x$ and $y$ share a common ancestor in $N$. In particular, we show that for any connected graph $G$ with vertex set $X$ and edge clique cover $K$ in which there are no two distinct sets in $K$ with one a subset of the other, there is some network with $|K|$ roots and leaf set $X$ whose shared ancestry graph is $G$.

翻译：设$X$为有限集，$|X|\geq 2$。网络$N$是定义在$X$上的有向无环连通图，其叶集为$X$，且$N$中每个根节点的出度至少为2，并且$N$中不存在入度和出度均为1的顶点；若$N$的底层无根无向图是一棵树，则称$N$为树状网络。网络在进化生物学中具有重要意义，例如用于表示一组物种$X$的进化历史，这些物种的祖先在过去曾发生过基因交换。对于任意符号集$M$，定义映射$d:{X \choose 2} \to M \cup \{\odot\}$为符号树状映射，若存在树状网络$N$，其中出度大于等于2的顶点被标记为$M$中的元素，且对任意$\{x,y\} \in {X \choose 2}$，$d(\{x,y\})$等于$x$和$y$在$N$中最近公共祖先的标记（若存在），否则为$\odot$。符号树状映射的重要实例包括符号超度量，该概念出现在博弈论、系统发育学和余图论等领域。本文证明：映射$d:{X \choose 2} \to M \cup \{\odot\}$是符号树状映射当且仅当$d$满足特定的三点和四点条件，且以$X$为顶点集、以所有满足$d(\{x,y\}) \neq \odot$的边$\{x,y\} \in {X \choose 2}$构成的图是托勒密图。为此，我们引入并证明关于$X$上网络$N$的共享祖先图的关键定理：该图以$X$为顶点集，边集由所有满足$x$和$y$在$N$中存在共同祖先的$\{x,y\} \in {X \choose 2}$组成。特别地，我们证明：对任意以$X$为顶点集的连通图$G$及其边团覆盖$K$（其中$K$中任意两个不同集合互不包含），存在一个具有$|K|$个根和叶集$X$的网络$N$，其共享祖先图恰为$G$。