  Relatively little is known about the discrete horospheres in hyperbolic
groups, even in simple settings. In this paper we work with hyperbolic
one-ended right-angled Coxeter groups and describe two graph structures that
mimic the intrinsic metric on a classical horosphere: the Rips graph and the
divergence graph (the latter due to Cohen, Goodman-Strauss, and Rieck). We
develop, analyze, and implement algorithms based on finite-state machines that
draw large finite portions of these graphs, and deduce various geometric
corollaries about the path metrics induced by these graph structures.
