Shared ancestry graphs and symbolic arboreal maps

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 2 ) → M ∪ {⊙ } is a symbolic arboreal map if there exists some arboreal network N whose vertices with outdegree 2 or more are labeled by elements in M and so that d({ x, y} ), { x, y} ∈ ( X 2 ) , is equal to the label of the least common ancestor of x and y in N if this exists, and ⊙ otherwise. 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 2 ) → M ∪ { ⊙} 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} ∈ ( X 2 ) with d({ x, y} ) not =odot is Ptolemaic (i.e., its shortest path distance satisfies Ptolemy's inequality). 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} ∈ ( X 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.