An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

Abstract

Let P be a set of n≥3 points in general position in the plane. The edge disjointness graph D(P) of P is the graph whose vertices are the n2 closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this paper we show that the connectivity of D(P) is at most 7n218+Θ(n), and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of D(Qn), where Qn denotes an n-point set that is almost 3-symmetric.

Keywords

• disjointness graph of segments
• rectilinear local crossing number
• 3-symmetry
• Menger’s theorem
• Hall’s theorem