Vietoris-Rips Complexes of Split-Decomposable Spaces

Split-metric decompositions are an important tool in the theory of phylogenetics, particularly because of the link between the tight span and the class of totally decomposable spaces, a generalization of metric trees whose decomposition does not have a ``prime'' component. Their close relationship with trees makes totally decomposable spaces attractive in the search for spaces whose persistent homology can be computed efficiently. We study the subclass of circular decomposable spaces, finite metrics that resemble subsets of $\mathbb{S}^1$ and can be recognized in quadratic time. We give an $O(n^2)$ characterization of the circular decomposable spaces whose Vietoris-Rips complexes are cyclic for all distance parameters, and compute their homotopy type using well-known results on $\mathbb{S}^1$. We extend this result to a recursive formula that computes the homology of certain circular decomposable spaces that fail the previous characterization. Going beyond totally decomposable spaces, we identify an $O(n^3)$ decomposition of $\mathrm{VR}_r(X)$ in terms of the blocks of the tight span of $X$, and use it to induce a direct-sum decomposition of the homology of $\mathrm{VR}_r(X)$.