$r$-indexing Wheeler graphs

Let G be a Wheeler graph and r be the number of runs in a Burrows-Wheeler Transform of G, and suppose G can be decomposed into υ edge-disjoint directed paths whose internal vertices each have inand out-degree exactly 1. We show how to store G in O(r + υ) space such that later, given a pattern P , in O(|P | log log |G|) time we can count the vertices of G reachable by directed paths labelled P , and then report those vertices in O(log log |G|) time per vertex.