Arrangements of pseudocircles and circles

Abstract

An arrangement of pseudocircles is a finite collection of Jordan curves in the plane with the additional properties that (i) no three of the curves meet in a point; (ii) every two curves meet in at most two points; and (iii) if two curves meet in a point p, then they cross at p.

We say that two arrangements b = (c₁,…, c _n ), D = (d₁,…, d _n ) are equivalent if there is a homeomorphism φ of the plane onto itself such that φ[c _i ] = d _i for all 1 ≤ i ≤ n. Linhart and Ortner (2005) gave an example of an arrangement of five pseudocircles that is not equivalent to an arrangement of circles, and conjectured that every arrangement of at most four pseudocircles is equivalent to an arrangement of circles. We prove their conjecture.

We consider two related recognition problems. The first is the problem of deciding, given a pseudocircle arrangement, whether it is equivalent to an arrangement of circles. The second is deciding, given a pseudocircle arrangement, whether it is equivalent to an arrangement of convex pseudocircles. We prove that both problems are NP-hard, answering questions of Bultena, Grünbaum and Ruskey (1998) and of Linhart and Ortner (2008).

We also give an example of a collection of convex pseudocircles with the property that its intersection graph (i.e. the graph with one vertex for each pseudocircle and an edge between two vertices if and only if the corresponding pseudocircles intersect) cannot be realized as the intersection graph of a collection of circles. This disproves a folklore conjecture communicated to us by Pyatkin.

Part of the work in this paper was done while this author was at Durham University, supported by EPSRC grant EP/G066604/1. He is now supported by a VENI grant from Netherlands Organisation for Scientific Research (NWO).

Part of the work in this paper was done while this author was supported by a VENI grant from Netherlands Organisation for Scientific Research (NWO).