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 δn, 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 / 3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(nlogn). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2 / 3}log 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.