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 (Ergodic Theory and Dynamical Systems 42(9):2740-2783, 2022)). 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.
