Clique-Based Separators for Geometric Intersection Graphs

Abstract

AbstractLet F be a set of n objects in the plane and let $$\mathcal {G}^{\times }(F)$$ G × ( F ) be its intersection graph. A balanced clique-based separator of $$\mathcal {G}^{\times }(F)$$ G × ( F ) is a set $$\mathcal {\mathcal {S}}$$ S consisting of cliques whose removal partitions $$\mathcal {G}^{\times }(F)$$ G × ( F ) into components of size at most $$\delta n$$ δ n , for some fixed constant $$\delta <1$$ δ < 1 . The weight of a clique-based separator is defined as $$\sum _{C\in \mathcal {\mathcal {S}}}\log (|C|+1)$$ ∑ C ∈ S log ( | C | + 1 ) . Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then $$\mathcal {G}^{\times }(F)$$ G × ( F ) admits a balanced clique-based separator of weight $$O(\sqrt{n})$$ O ( n ) . We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight $$O(\sqrt{n})$$ O ( n ) , which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight $$O(n^{2/3}\log n)$$ 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 $$O(\sqrt{n}\log n)$$ O ( n log n ) . (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight $$O(n^{2/3}\log n)$$ 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 $$O(\sqrt{n}+r\log (n/r))$$ O ( n + r log ( 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.