On Arrangements of Orthogonal Circles

Abstract

In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NPhard to recognize orthogonal unit circle intersection graphs.

Arrangements of orthogonal circles with many intersections

“…We begin by stating a few properties of arrangements of orthogonal circles. The first lemma was proven by Chaplick et al [4]. Proof.…”

“…We apply a Möbius transformation that maps A to a line. Note that such a transformation is conformal and thus maintains the angles; see Figure The following lemma is again taken from Chaplick et al [4,Lemma 5]. The "Moreover"-part is not explicitly written down, but it is apparent from the construction given in its proof.…”

“…For every n ≥ 6 for which n mod 5 = 1 the arrangement B (n−1)/5,5 with only the innermost hub circle is nonnested and its n-vertex intersection graph has 3n − 8 edges. Chaplick et al [4] investigated the maximal number of triangular cells in an orthogonal circle arrangement. They proved an upper bound of 4n and gave a lower bound of 2n triangular cells, which they later improved to 3n − 3.…”

“…Arrangements of orthogonal circles and their intersection graphs were recently introduced by Chaplick et al [4]. Here it was shown that the intersection graph contains at most 7n edges.…”

“…Every arrangement of orthogonal circles with the same radius can be turned into a unit circle packing by shrinking the circle size by a factor of √ 2/2, but there are unit disk contact graphs that are not intersection graphs of an arrangement of orthogonal circles [4].…”