  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. We use this connection to
study the Vietoris-Rips complexes of totally decomposable spaces. In
particular, we characterize the homotopy type of the Vietoris-Rips complex of
circular decomposable spaces whose metric is monotone. We extend this result to
compute the homology of certain non-monotone circular decomposable spaces. We
also use the block decomposition of the tight span of a totally decomposable
metric to induce a decomposition of the VR complex of a totally decomposable
metric.
