Topological Drawings meet Classical Theorems from Convex Geometry∗ Helena Bergold1, Stefan Felsner2, Manfred Scheucher2, Felix Schröder2, and Raphael Steiner2 1 Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Germany, {helena.bergold}@fernuni-hagen.de 2 Institut für Mathematik, Technische Universität Berlin, Germany, {felsner,scheucher,fschroed,steiner}@math.tu-berlin.de 1 Abstract 2 In this article, we discuss classical theorems from Convex Geometry in the context of simple 3 topological drawings of the complete graph Kn. In a simple topological drawing, any two edges 4 share at most one point: either a common vertex or a point where they cross. 5 We present generalizations of Kirchberger’s Theorem and the Erdős–Szekeres Theorem, a 6 family of simple topological drawings with arbitrarily large Helly number, a new proof of the gen- 7 eralized Carathéodory’s Theorem, and discuss further classical theorems from Convex Geometry 8 in the context of simple topological drawings. Lines 175 9 1 Introduction 10 In a topological drawing D of the complete graph Kn, vertices are mapped to points in the 11 plane and edges are mapped to simple curves connecting the corresponding end-points such 12 that every pair of edges has at most one common point, which is either a common endpoint 13 or a crossing; see Figure 1. Note that several edges may cross in a single point. A topological 14 drawing is called straight-line if all edges are drawn as line segments, and pseudolinear if all 15 arcs of the drawing can be extended to bi-infinite curves such that any two of these curves 16 cross at most once (the family of curves is an arrangement of pseudolines). 17 Figure 1 Forbidden patterns in topological drawings: self-crossings, double-crossings, touchings, 18 and crossings of adjacent edges. 19 1.1 Our Results 20 In Section 2, we present a new combinatorial generalization of topological drawings – which 21 we name generalized signotopes. They allow us to prove a generalization of Kirchberger’s 22 Theorem in Section 3. We present a family of topological drawings with arbitrarily large ∗ S. Felsner and M. Scheucher were supported by DFG Grant FE 340/12-1. R. Steiner was supported by DFG-GRK 2434. We thank Winfried Hochstättler for valuable discussions and helpful comments. 36th European Workshop on Computational Geometry, Würzburg, Germany, March 16–18, 2020. This is an extended abstract of a presentation given at EuroCG’20. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear eventually in more final form at a conference with formal proceedings and/or in a journal. 21:2 Topological Drawings meet Classical Theorems from Convex Geometry 23 Helly number in Section 4, and, in Section 5, we discuss generalizations of Carathéodory’s 24 Theorem, Erdős–Szekeres Theorem, and colorful variants of the classical theorems. 25 In the full version, we also describe the SAT model, which we used to test hypotheses 26 about simple topological drawings, and give a more detailed analysis of the named theorems 27 in terms of the convexity hierarchy introduced by Arroyo, McQuillan, Richter, and Salazar [1]. 28 In particular, some of the theorems turn out to generalize to pseudolinear drawings but not 29 to pseudocircular drawings, i.e., topological drawings where all arcs can be simultaneously 30 extended to pseudocircles such that any two do not touch and cross at most twice. 31 2 Preliminaries 32 Given a topological drawing D, we call the induced subdrawing of three vertices a triangle. 33 Note that the edges of a triangle in a topological drawing do not cross. The removal of a 34 triangle separates the plane into two connected components. A point p is in the interior 35 of a triangle or more generally of a topological drawing D if p is in a bounded connected 2 36 component of R − D. 37 In a topological drawing, we assign an orientation χ(abc) ∈ {+, −} to each ordered triple 38 abc of vertices. The sign χ(abc) indicates whether we go counterclockwise or clockwise around 39 the triangle when visiting the vertices a, b, c in this order. 40 In a straight-line drawing of Kn the underlying point set S = {s1, . , sn} is in general 41 position (no three points lie on a line). If the points are sorted from left to right, then 42 for every 4-tuple si, sj, sk, sl with i < j < k < l the sequence χ(ijk), χ(ijl), χ(ikl), χ(jkl) 43 (index-triples in lexicographic order) is monotone, i.e., there is at most one sign-change. A [n] 44 signotope is a mapping χ : 3 → {+, −} with the above monotonicity property. Signotopes 45 are in bijection with Euclidean pseudoline arrangements and can be used to characterize 46 pseudolinear drawings [8, 3]. 47 Let us now consider topological drawings of the complete graph. There are two types 48 of drawings of K4 on the sphere: type I has a crossing and type II has no crossing. Type I 49 can be drawn in two different ways in the plane: in type Ia the crossing is only incident to 50 bounded faces and in type Ib the crossing lies on the outer face; see Figure 2. 4 3 4 3 4 123+ 123+ 123+ 124+ 124+ 124+ 134+ 134− 3 134+ 234+ 234− 234− 1 2 1 2 1 2 type Ia type Ib type II 51 Figure 2 The three types of topological drawings of K4 in the plane. 52 A drawing of K4 with vertices a, b, c, d can be characterized in terms of the sequence of 53 orientations χ(abc), χ(abd), χ(acd), χ(bcd). The drawing is 54 of type Ia or type Ib iff the sequence is + + ++, + + −−, + − −+, − + +−, − − ++, or 55 − − −−; and 56 of type II iff the number of +’s (and −’s respectively) in the sequence is odd. H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner 21:3 57 We conclude that there are at most two sign-changes in the sequence and that, moreover, 58 any such sequence is in fact induced by a topological drawing of K4. Allowing up to two 59 sign-changes is equivalent to forbidding the two patterns + − +− and − + −+. 60 If a mapping χ :[n]3 → {+, −} is alternating, i.e., χ(iσ(1), iσ(2), iσ(3)) = sgn(σ) · 61 χ(i1, i2, i3), and χ avoids the two patterns on sorted indices, i.e., χ(ijk), χ(ijl), χ(ikl), χ(jkl) 62 has at most two sign-changes for i < j < k < l, then it avoids the two patterns on 63 χ(abc), χ(abd), χ(acd), χ(bcd) for any pairwise different a, b, c, d ∈ [n]. We refer to this as the 64 symmetry property of the forbidden patterns. 65 The symmetry property allows us to define generalized signotopes as alternating mappings 66 χ :[n]3 → {+, −} with at most two sign-changes on χ(abc), χ(abd), χ(acd), χ(bcd) for any 67 pairwise different a, b, c, d ∈ [n]. We remark that generalized signotopes can be seen as a 68 proper generalization of topological drawings – details are deferred to the full version [6]. 69 3 Kirchberger’s Theorem d 70 Two point sets A, B ⊆ R are called separable if there exists a hyperplane H separating 71 them. It is well-known that if two sets A, B are separable they can also be separated by 72 a hyperplane H containing one point of A and B. Kirchberger’s Theorem (see [11] or [5]) d 73 asserts that two finite point sets A, B ⊆ R are separable if and only if for every C ⊆ A ∪ B 74 with |C| = d + 2, C ∩ A and C ∩ B are separable. 75 We prove a generalization of the 2-dimensional version of Kirchberger’s Theorem in 76 the setting of generalized signotopes, where two sets A, B ⊆ [n] are separable if there exist 77 i, j ∈ A∪B such that χ(i, j, x) = + for all x ∈ A\{i, j} and χ(i, j, x) = − for all x ∈ B\{i, j}. 78 In this case we say that ij separates A from B and write χ(i, j, A) = + and χ(i, j, B) = −. 79 Moreover, if we can find i ∈ A and j ∈ B, we say that A and B are strongly separable. 80 I Theorem 3.1 (Kirchberger’s Theorem for Generalized Signotopes). Let χ :[n]3 → {+, −} be 81 a generalized signotope, and A, B ⊆ [n]. If for every C ⊆ A ∪ B with |C| = 4, the sets A ∩ C 82 and B ∩ C are (strongly) separable, then A and B are (strongly) separable. 83 Proof. First note that it is sufficient to prove the theorem for strongly separable since 84 all 4-tuples which are weakly separable are also strongly separable. For more details, see 85 Appendix A. Hence in the following we always assume that the 4-tuples are strongly separable. 86 To prove the claim we consider a counterexample (χ, A, B) minimizing the size of the 87 smaller of the two sets. By symmetry we may assume |A| ≤ |B|. First we consider the cases 88 |A| = 1, 2, 3 individually and then the case |A| ≥ 4. 0 89 Let A = {a}, we need to find b ∈ B such that χ(a, b, B) = −. Let B be a maximal subset 0 0 90 of B such that B can be separated from {a}, and let b ∈ B be such that χ(a, b, B ) = −.