Clique-Based Separators for Geometric Intersection Graphs
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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. (SICOMP 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. Map graphs admit a balanced clique-based separator of weight O(√(n)), which is tight in the worst case. 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). Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^2/3log n). 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.
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