Shared Ancestry Graphs and Symbolic Arboreal Maps

A network N on a finite set X, |X|≥ 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 two or more are labelled 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 ⊙ 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 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}) ≠⊙ 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}∈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.