Tightness of balanced clique-based separator bounds in hyperbolic uniform disk graphs

Determine whether the O((1 + 1/r) log n)-clique balanced separator bound for hyperbolic uniform disk graphs HUDG(r)—the intersection graphs of equal-radius r disks in the hyperbolic plane—is tight for all radii r. Specifically, ascertain if the logarithmic factor can be reduced for larger (super-constant) radii r, or prove matching lower bounds that confirm tightness beyond the constant-r regime where tightness is already known via regular hyperbolic tilings with logarithmic treewidth.

Background

The paper proves that any n-vertex hyperbolic uniform disk graph with disk radius r admits a balanced separator covered by O((1 + 1/r) log n) cliques, computable in O(n log n) time. This generalizes known hyperbolic separator results and yields algorithmic consequences for Independent Set and related problems.

For constant radii r, the authors note the existence of regular hyperbolic tilings whose associated graphs have constant clique number and logarithmic treewidth, aligning with the proven O(log n)-clique separator, suggesting asymptotic tightness in that regime.

However, for larger radii (e.g., r in Ω(log n)), the structure may simplify further, and it is not established whether the logarithmic factor in the separator bound can be improved or whether matching lower bounds enforce tightness across all r.