A graph-theoretical invariant of topological spaces

Given a topological space TT and a finite subset T0T0 of TT, we associate two graphs with TT and T0T0 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 TT may model a full “orbit” of a dynamical branching process, and T0T0 the set of its branching points.