Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let 𝒢^×(F) be its intersection graph. A balanced clique-based separator of 𝒢^×(F) is a set 𝒮 consisting of cliques whose removal partitions 𝒢^×(F) into components of size at most δ n , for some fixed constant δ <1 . The weight of a clique-based separator is defined as ∑ _C∈𝒮log (|C|+1) . Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then 𝒢^×(F) admits a balanced clique-based separator of weight O(√(n)) . We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(√(n)) , which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^2/3log n) . If the pseudo-disks are polygonal and of total complexity O ( n ) then the weight of the separator improves to O(√(n)log n) . (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^2/3log n) . (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√(n)+rlog (n/r)) , which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover ), for Feedback Vertex Set , and for q - Coloring for constant q in these graph classes.