A time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the NNI graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by applying the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Furthermore, we design an efficient algorithm for constructing these graphs. Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of open problems of prominent importance for those applications.
In this talk, I will present these results supported by all necessary background. The detailed exposition can be accessed at http://biorxiv.org/content/early/2016/07/12/063362 and a nice and informal introduction at http://matsen.fredhutch.org/general/2016/07/11/discrete-time-tree.html