Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of \(\mathbb {R}^2\) and let \(\mathcal {D}=\{D_1,\ldots,D_n\}\) be a set of geodesic disks with respect to the metric d. We prove that \(\mathcal {G}^{\times }(\mathcal {D})\) , the intersection graph of the disks in \(\mathcal {D}\) , has a clique-based separator consisting of \(O(n^{3/4+\varepsilon })\) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time \(2^{O(n^{3/4+\varepsilon })}\) , assuming the boundaries of the disks \(D_i\) can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses \(O(n^{7/4+\varepsilon })\) storage and can report the hop distance between any two nodes in \(\mathcal {G}^{\times }(\mathcal {D})\) in \(O(n^{3/4+\varepsilon })\) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.
