Near-Optimal Separators in String Graphs

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of O(m √ logm) vertices, and they conjectured that the bound of O( √ m) actually holds. We obtain separators with O( √ m logm) vertices. Let G = (V,E) be a graph with n vertices. A separator in G is a set S ⊆ V of vertices such that there is a partition V = V1 ∪ V2 ∪ S with |V1|, |V2| ≤ 23n and no edges connecting V1 to V2. The graph G is a string graph if it is an intersection graph of curves in the plane, i.e., if there is a system (γv : v ∈ V ) of curves (continuous arcs) such that γu ∩ γv 6= ∅ iff {u, v} ∈ E(G) or u = v. Fox and Pach [FP10] proved that every string graph has a separator with O(m3/4 √ logm) vertices, where m is the number of edges of G. We should mention that they actually proved the result for the weighted case, where each vertex v ∈V has a positive real weight, and the size of the components of G\S is measured by the sum of vertex weights (while the size of S is still measured as the number of vertices). Our result can also be extended to the weighted case, either by deriving it from the unweighted case along the lines of [FP10], or by using appropriate vertex-weighted versions (available in the cited sources) of the tools used in the proof. However, for simplicity, we stick to the unweighted case in this note. Pach and Fox conjectured that string graphs actually have separators of size O( √ m ) (which, if true, would be asymptotically optimal in the worst case). Earlier, in [FP08], they proved some special cases of this conjecture: most notably, if every two curves γu, γv in the string representation intersect in at most k points, where k is a constant. As they kindly informed me in February 2013, they also have an (unpublished) proof of existence of separators of size O( √ n ) in string graphs with maximum degree bounded by a constant. Here we obtain the following result. Theorem 1. Every string graph G with m ≥ 2 edges has a separator with O(√m logm) vertices. ∗Supported by the ERC Advanced Grant No. 267165 and by GRADR Eurogiga GIG/11/E023.