A graph-theoretical invariant of topological spaces

Given a topological space T and a finite subset T 0 of T , we associate two graphs with T and T 0 that, under rather mild conditions, turn out to be a block graph and a tree, respectively. This construction is of interest, e.g., in the context of phylogenetic analysis where T may model a full “orbit” of a dynamical branching process, and T 0 the set of its branching points.