Structure and Independence in Hyperbolic Uniform Disk Graphs

We consider intersection graphs of disks of radius r in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of r, where very small r corresponds to an almost-Euclidean setting and r ∈ Ω(log n) corresponds to a firmly hyperbolic setting. We observe that larger values of r create simpler graph classes, at least in terms of separators and the computational complexity of the {Independent Set} problem. First, we show that intersection graphs of disks of radius r in the hyperbolic plane can be separated with O((1+1/r)log n) cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set S of n points with pairwise distance at least $2r$ in the hyperbolic plane the corresponding Delaunay complex has outerplanarity 1+O(log n/r), which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that {Independent Set} can be solved in nO(1+log n/r) time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, {Independent Set} is polynomial-time solvable in the firmly hyperbolic setting of r∈ Ω(log n). Finally, in the case when the disks have ply (depth) at most , we give a PTAS for {Maximum Independent Set} that has only quasi-polynomial dependence on 1/ε and . Our PTAS is a further generalization of our exact algorithm.