Topological Drawings meet Classical Theorems from Convex ∗

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 : 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- 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 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 ) = −. 0 ∗ 0 91 Suppose that B 6= B, then there is a b ∈ B\B with

∗ 92 χ(a, b, b ) = +. (1)

0 0 ∗ 93 By maximality of B , the sets {a} and B ∪ {b } cannot be separated. Hence, we have

∗ 0 94 χ(a, b , b ) = + (2)

0 0 0 0 0 95 for some b ∈ B . Since χ is alternating (1) and (2) together imply b 6= b. Since b ∈ B we 0 96 have χ(a, b, b ) = −. From this together with (1) and (2) it follows that the four element set 0 ∗ 0 97 {a, b, b , b } has no separator. This is a contradiction, whence B = B.

E u r o C G ’ 2 0 21:4 Topological Drawings meet Classical Theorems from Convex Geometry

98 As a consequence we obtain:

99 Every one-element set {a} with a ∈ A can be separated from B. Since χ is alternating

100 there can be at most one b(a) ∈ B such that χ(a, b(a),B) = −.

101 Now we look at the case where A = {a1, a2}. Let bi = b(ai), if b1 = b2, or χ(a1, b1, a2) = 102 +, or χ(a2, b2, a1) = + we have a separator for A and B. So assume that b1 6= b2, 103 and χ(a1, b1, a2) = −, and χ(a2, b2, a1) = −. Since χ is alternating we also know that 104 χ(a1, b2, b1) = + and χ(a2, b1, b2) = +. Together these four signs show that {a1, b1, a2, b2} is

105 not separable, a contradiction.

106 The case |A| = 3 works similarly but is more technical; see Appendix B.

107 Now we consider the remaining case where (χ, A, B) is a minimal counterexample with

108 4 ≤ |A| ≤ |B|. Every one-element set {a} can be separated from B. Since χ is alternating

109 there can be at most one b(a) ∈ B such that χ(a, b(a),B) = −. ∗ ∗ 0 110 Let a ∈ A. By minimality of (χ, A, B), A\{a } is separable from B. Let a ∈ A and 0 111 b ∈ B such that χ(a, b, A ) = + and χ(a, b, B) = −. Hence it is

∗ 112 χ(a, b, a ) = −. (3)

∗ ∗ ∗ ∗ 0 ∗ 0 113 Let b = b(a ), i.e., χ(a , b ,B) = −. There is some a ∈ A\{a } = A such that

∗ ∗ 0 114 χ(a , b , a ) = −. (4)

0 ∗ ∗ ∗ 115 If a = a, then b 6= b because of (3) and (4). From χ(a, b, B) = −, χ(a , b ,B) = −, (3), ∗ ∗ 116 and (4) it follows that the 4-element set {a, b, a , b } has no separation. The contradiction 0 117 shows a 6= a. 0 0 0 0 0 0 0 0 118 Let b = b(a ). If b = b , then a ∈ A implies χ(a, b, a ) = + which yields χ(a , b , a) = −. 0 0 0 119 If b 6= b we look at the four elements a, b, a , b , and have:

0 0 0 0 0 0 120 χ(a , b , a) =? , χ(a , b , b) = − , χ(a , a, b) = + , χ(b , a, b) = −

0 0 0 0 121 To avoid the forbidden pattern for a b ab we must have χ(a , b , a) = −. 0 0 122 Hence, regardless whether b = b or b 6= b we have

0 0 123 χ(a , b , a) = − . (5)

0 0 ∗ ∗ 124 Since |A| ≥ 4, we know by the minimality of (χ, A, B) that the set {a, b, a , b , a , b },

125 which has 3 elements of A and at least 4 elements in total, is separable. It follows from 0 0 ∗ ∗ 0 0 126 χ(a, b, B) = χ(a , b ,B) = χ(a , b ,B) = + that the only possible separators are ab, a b , ∗ ∗ 127 and a b . They, however, do not separate because of (3), (5), and (4) respectively. This is 128 impossible, hence, there is no counterexample. J

129 4 Helly’s Theorem

d 130 Helly’s Theorem asserts that the intersection of n convex sets S1,...,Sn in R is non-empty 131 if the intersection of every d + 1 of these sets in non-empty. In other words, the Helly number d 132 of a family of n convex sets in R is at most d + 1. The result of Goodman and Pollack [9] 133 (see also [2]) shows that Helly’s Theorem holds for pseudoconfigurations of points in two

134 , and thus for pseudolinear drawings.

135 In the more general setting of topological drawings, we investigate the intersection

136 properties of triangles. We show that Helly’s Theorem does not generalize to topological

137 drawings by constructing topological drawings with arbitrarily large Helly number. H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner 21:5

138 I Theorem 4.1. For every odd integer n, there exists a topological drawing of K3n with Helly 139 number at least n, i.e., there are n triangles such that any n − 1 have a common interior,

140 but not all n have a common interior.

141 Sketch of the proof. The basic idea of the construction is to draw n triangles on the cylinder

142 with the property that any n − 1 triangles have a common interior, while there is no common

143 intersection of all n triangles; the left-hand side of Figure 3 gives an illustration. Such a

144 cylindrical drawing can clearly be drawn in the plane as depicted on the right-hand side

145 of Figure 3. The technical part, however, is to complete such a drawing of n triangles to a

146 topological drawing of the complete graph K3n. Technical details are deferred to Appendix C.

147 Figure 3 An illustration of a topological drawing of K3n with Helly number n.

2 148 For the construction, we identify the cylinder surface with the real plane R . As illustrated 149 in Figure 4, we place the vertices of K3n on 3 different layers.

y = n + 1

y = n

y = 0

150 Figure 4 Placement of the vertices and drawing edges between different layers.

151 The edges between different layers are drawn as straight-line segments (see Figure 4).

152 Edges between two vertices of the same layer are drawn as circular arcs above the top layer,

153 below the bottom layer and in a sufficiently small range above the middle layer (see Figure 5). 154 The obtained drawing is topological (see Appendix C), which concludes the proof. J

156 5 Further Results

157 Carathéodory’s Theorem asserts that, if a point x lies in the of a point set P d 158 in R , then x lies in the convex hull of at most d + 1 points of P . A more general version 159 of Carathéodory’s Theorem in the plane is by Balko, Fulek, and Kynčl, who provided a

160 generalization to topological drawings [3, Lemma 4.7]. In Appendix D, we present a new

161 proof for their theorem.

E u r o C G ’ 2 0 21:6 Topological Drawings meet Classical Theorems from Convex Geometry

y = n + ε0

y = n

y = n − ε0

155 Figure 5 Edges between middle-layer vertices are drawn as very flat circular arcs.

162 I Theorem 5.1 (Carathéodory for Topological Drawings [3]). Let D be a topological drawing 2 163 of Kn and let x ∈ R be a point in the interior of D. Then there is a triangle in D which 164 contains x in its interior.

165 In the full version, we also study colorful variants of the theorems. For example it turned

166 out that Barany’s Colorful Carathéodory Theorem [4] holds up to pseudolinear drawings [10]

167 but not for pseudocircular drawings (see Figure 6).

0 1 2 2 1 0 0 1 2

168 Figure 6 A circular drawing of K9 violating the condition of Colorful Carathéodory Theorem.

169 The classical Erdős–Szekeres Theorem [7] asserts that every straight-line drawing of Kn 170 contains a crossing-maximal subdrawing of size k = Ω(log n), that is, a subdrawing of Kk k 171 with 4 crossings. Pach, Solymosi, and Tóth [12] generalized this result by showing that 1/8 172 every topological drawing of Kn has a crossing-maximal subdrawing of size k = Ω(log n).

173 In the full version [6], we show that generalized signotopes on 4 and 5 elements correspond

174 to drawings of K4 and K5, respectively, and a simple Ramsey–type argument then shows

175 that a variant of the Erdős–Szekeres Theorem even applies to generalized signotopes.

176 References

177 1 A. Arroyo, D. McQuillan, R. B. Richter, and G. Salazar. Convex drawings of the complete

178 graph: topology meets geometry. arXiv:1712.06380, 2017.

179 2 A. Bachem and A. Wanka. Separation theorems for oriented matroids. Discrete Mathe-

180 matics, 70(3):303–310, 1988.

181 3 M. Balko, R. Fulek, and J. Kynčl. Crossing Numbers and Combinatorial Characterization

182 of Monotone Drawings of Kn. Discrete & Computational Geometry, 53(1):107–143, 2015. H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner 21:7

183 4 I. Bárány. A generalization of Carathéodory’s Theorem. Discrete , 40(2):141–

184 152, 1982.

185 5 A. Barvinok. A course in convexity, volume 54 of Graduate Studies in Mathematics. Amer-

186 ican Mathematical Society, 2002.

187 6 H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner. Topological Drawings

188 meet Classical Theorems from Convex Geometry. arXiv:TODO, 2019+.

189 7 P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica,

190 2:463–470, 1935.

191 8 S. Felsner and H. Weil. Sweeps, Arrangements and Signotopes. Discrete Applied Mathe-

192 matics, 109(1):67–94, 2001. 2 193 9 J. E. Goodman and R. Pollack. Helly-type theorems for pseudoline arrangements in P .

194 Journal of Combinatorial Theory, Series A, 32(1):1–19, 1982.

195 10 A. F. Holmsen. The intersection of a matroid and an oriented matroid. Advances in

196 Mathematics, 290:1–14, 02 2016.

197 11 P. Kirchberger. Über Tchebychefsche Annäherungsmethoden. Mathematische Annalen,

198 57:509–540, 1903.

199 12 J. Pach, J. Solymosi, and G. Tóth. Unavoidable configurations in complete topological

200 graphs. Discrete & Computational Geometry, 30(2):311–320, 2003.

E u r o C G ’ 2 0 21:8 Topological Drawings meet Classical Theorems from Convex Geometry

201 A Weak Separable implies Strong Separable for 4-tuples

202 Tables 1 and 2 show that, in every weakly-separable generalized signotopes on {a, b1, b2, b3} 203 and {a1, a2, b1, b2}, respectively, there is a strong separator.

χ(a, b1, b2) χ(a, b1, b3) χ(a, b2, b3) χ(b1, b2, b3) list of separators + + + + ab3, b1a, b1b3 + + + − ab3, b1a, b1b2, b2b3 + + − + ab2, b1a, b1b3, b3b2 + + − − ab2, b1a, b1b2 + − + + (no separators) + − − + ab2, b3a, b3b2 + − − − ab2, b1b2, b3a, b3b1 − + + + ab3, b1b3, b2a, b2b1 − + + − ab3, b2a, b2b3 − + − − (no separators) − − + + ab1, b2a, b2b1 − − + − ab1, b2a, b2b3, b3b1 − − − + ab1, b2b1, b3a, b3b2 − − − − ab1, b3a, b3b1

204 Table 1 Separators for generalized signotopes on {a1, b1, b2, b3}. Strong separators are underlined.

χ(a1a2b1) χ(a1a2b2) χ(a1b1b2) χ(a2b1b2) list of separators + + + + a2a1, a2b2, b1a1, b1b2 + + + − a2a1, a2b1, b1a1 + + − + a2a1, a2b2, b2a1 + + − − a2a1, a2b1, b2a1, b2b1 + − + + a1b2, b1a1, b1b2 + − − + (no separators) + − − − a2b1, b2a2, b2b1 − + + + a2b2, b1a2, b1b2 − + + − (no separators) − + − − a1b1, b2a1, b2b1 − − + + a1a2, a1b2, b1a2, b1b2 − − + − a1a2, a1b2, b2a2 − − − + a1a2, a1b1, b1a2 − − − − a1a2, a1b1, b2a2, b2b1

205 Table 2 Separators for generalized signotopes on {a1, a2, b1, b2}. Strong separators are underlined.

206 B Proof of the case |A| = 3 in Theorem 3.1

207 Let A = {a1, a2, a3}. Suppose that A is not separable from B. Let bi = b(ai), i.e., 208 χ(ai, bi,B) = − for i = 1, 2, 3. For i, j ∈ {1, 2, 3}, i 6= j we define sij = χ(ai, bi, aj). 209 If sij = sik = + for j 6= k, then ai, bi separates A from B, hence, at least one of sij and 210 sik is −. H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner 21:9

211 If sij = sji = −, then since χ is alternating bi 6= bj and there is no separation for 212 {ai, bi, aj, bj}. Hence, at least one of sij and sji is +. 213 These two conditions imply that we can relabel the elements of A such that s12 = s23 = 214 s31 = + and s13 = s21 = s32 = −. Assuming bi = bj = b we find that there is no separation 215 for {a1, a2, a3, b}. Hence, the b1, b2, b3 are pairwise distinct. 216 Furthermore {a1, a2, a3, b3} must have a separator. From s32 = − and s31 = +, i.e., 217 χ(a1, b3, a3) = −, we conclude that the separator is a2b3. In particular,

218 χ(a2, b3, a1) = +. (6)

219 The forbidden pattern + − +− for generalized signototpes together with

220 s12 = χ(a1, b1, a2) = + , s13 = χ(a1, b1, b3) = − , χ(a1, a2, b3) = + (see (6))

221 implies

222 χ(b1, a2, b3) = +. (7)

223 The set {b1, a2, a3, b3} needs a separator, the candidate pair a3, b1 is made impossible by 224 χ(a3, b3, b1) = −, a3, b3 is made impossible by s32 = −, and a2, b3 is made impossible by (7). 225 Hence a2, b1 is the separator and, in particular, it holds

226 χ(a2, b1, a3) = +. (8)

227 But now the set {a1, a2, a3, b1} has no separator. a1, b1 is not a separator because of s13 = −, 228 a2, b1 does not separate because χ(a2, b2, b1) = −, and (8) shows that a3b1 cannot separate

229 the set. This contradiction to the assumption proves the case |A| = 3.

230 C Proof of Theorem 4.1

2 231 For the construction, we identify the cylinder surface with the real plane R , where x- 232 coordinates are considered modulo n, that is, the points {(x + k · n, y): k ∈ Z} are identified 233 for every x, y ∈ R.

234 We place the vertices of K3n on 3 different layers:

235 bot-vertices on (k, 0) for k ∈ Z, 236 mid-vertices on (k − ε, n) for k ∈ Z (with ε sufficiently small positive constant), and 237 top-vertices on (k, n + 1) for k ∈ Z.

238 As illustrated in Figure 4, all edges between different layers are drawn as straight-line

239 segments on the cylinder, which have particular windings, i.e., tracing a planar

240 e from its top-most point (x0, y0) to its bottom-most point (x1, y1), we define the winding 241 w(e) as the difference x1 − x0.

242 The edges are drawn as follows:

243 top-mid edges: starting from a middle vertex, we connect to all vertices of the top layer

244 by a straight-line on the cylinder to the right. The winding of a such a top-mid edge

245 fulfills −(n − 1 + ε) ≤ w(e) ≤ −ε.

246 mid-bot edges: starting from a middle vertex, we connect to all vertices of the bottom

247 layer by a straight-line on the cylinder to the right. The winding of such a bot-mid-edge

248 fulfills ε ≤ w(e) ≤ n − 1 + ε.

E u r o C G ’ 2 0 21:10 Topological Drawings meet Classical Theorems from Convex Geometry

249 top-bot edges: starting from a top vertex, we connect to all vertices of the bottom

250 layer by a straight-line to the right. The winding of such a top-bottom edge fulfills

251 0 ≤ w(e) ≤ n − 1.

252 We will come to top-top, bot-bot, and mid-mid edges later; but first we show that

253 the edges drawn so far correspond to a topological drawing. The main observation is the

254 following:

255 I Lemma 3.1. If |w(e) − w(f)| < n, then the two segments e and f cross at most once.

256 Proof. Suppose that two segments e and f have two crossing points p and q on the cylinder

257 surface. We may assume w.l.o.g. that e and f start in p and end in q, and – by a simple

258 skewing argument – that e is vertical. Since e and f are not identically, f wraps around the 259 cylinder at least once, and the statement follows. J

260 It follows directly that each pair of top-mid, mid-bot, or top-bot edges, respectively, can

261 cross at most once. It is also clear by construction, that top-mid edges meet mid-bot edges

262 in at most one point.

263 Assuming – towards a contradiction – that there are two edges e and f which cross in

264 two or more points, then, we have one of the two following cases:

265 Case 1:

266 e is a top-bot edge and f is a top-mid edge. Since f entirely lies above y ≥ n, we look at

267 the pruned segment e˜ ⊆ e which starts at height y = n + 1 and y = n (see Figure 7). This |w(e)| n 268 segment has winding |w(e˜)| = n+1 < n+1 . Since |w(f)| < n − 1, we have |w(e˜) − w(f)| < n 269 and thus e˜ and f cannot cross twice. Hence, this case cannot occur.

y = n + 1 f e˜ y = n

e

270 Figure 7 A top-bot and a top-mid edge intersect at most once.

271 Case 2:

272 e is a top-bot edge and f is a mid-bot edge. We have −1 < w(e) < n−1 and −1 ≤ w(f) ≤ n−2,

273 and thus |w(e) − w(f)| < n. So two crossings are also impossible in this case.

274 This is a contradiction to the assumption, that there are edges which cross more than

275 once. Hence, (so far) the drawing is topological.

276 Remaining edges:

277 The top-top edges are drawn as follows: for a top vertex at (k, n + 1), we draw circular arcs n−1 278 to the top-vertices with x ∈ {k + 1, . . . , k + 2 } to the right, circular arcs to the top-vertices n−1 279 with x ∈ {k − 1, . . . , k − 2 } to the left so that the circular arcs fully lie above y ≥ n + 1. 280 Analogously, the bot-bot edges are drawn below y ≤ 0. H. Bergold, S. Felsner, M. Scheucher, F. Schröder, and R. Steiner 21:11

0 281 We can also draw mid-mid edges as circular arcs in a very flat strip between n − ε ≤ 0 0 282 y ≤ n + ε for a sufficiently small positive constant ε , we have to be a little more careful in

283 this case. It is crucial to observe that – in the mentioned strip – all other edges look almost

284 like vertical segments as depicted in Figure 5. Hence, the obtained drawing is topological.

285 This completes the proof of Theorem 4.1.

286 D Proof of Theorem 5.1

287 Suppose towards a contradiction that there is a pair (D, x) violating the claim. We choose

288 D minimal w.r.t. the number of vertices n.

289 Let a be a vertex of the drawing. If we removed all incident edges of a from D, then

290 by minimality of the example x would become a point of the outer face. Therefore, if we 0 291 remove the incident edges of a one by one, we find a last subdrawing D such that x is still 0 292 in a bounded face. Let ab be the edge such that in the drawing D − ab the point x is in the

293 outer face.

294 There is a simple path P connecting x to infinity, which does not cross any of the edges 0 0 295 in D − ab. By the choice of D , P has at least one crossing with ab. We choose P minimal

296 with respect to the number of crossings with ab.

297 We claim that P intersects ab exactly once. Suppose that P crosses ab more than once.

298 Then there is a lense C formed by P and ab, that is, two crossings between P and ab which

299 are consecutive on P and on ab; see Figure 8a. This lense is bounded by a simple closed

300 curve ∂C, which is composed of a subpath P1 of P and a subpath P2 of ab, each between

301 the consecutive crossings.

b b P2 x p P C x P c d P1

a a (a) (b)

302 Figure 8 (a) and (b) give an illustration of the proof of Theorem 5.1. shows a non-topological draw- 303 ing of K4 (where every induced subdrawing is topological) violating the conditions of Carathéodory’s 304 Theorem.

0 305 Now consider the path P from x to infinity which is obtained from P by replacing the 0 306 subpath P1 by a path P2 which is a close copy of P2 in the sense that it has the same crossing 307 pattern with all edges in D and the same topological properties, but is disjoint from ab. As P

308 was chosen minimal with respect to the number of crossings with ab, there has to be an edge 0 0 0 309 of the drawing D that intersects P2 (and by the choice of P2 also P2). By the properties 310 of a topological drawing, this edge has one of its endpoints, say c, inside the lense C and 0 311 one outside C. Observe that the edge bc in D must intersect ∂C. By the properties of a

312 topological drawing bc cannot intersect ab and by the choice of P it does not intersect P .

313 The contradiction shows that P crosses ab in a unique point p. 0 314 If a has another neighbor c in the drawing D then there is an edge connecting b to c in 0 315 the drawing D since all vertices are fully connected except the vertex a. The edges ac and

316 bc cannot cross P , so x is in the interior of the triangle abc, and the statement follows.

E u r o C G ’ 2 0 21:12 Topological Drawings meet Classical Theorems from Convex Geometry

0 317 In the remaining case, we assume deg(a) = 1 in D . As x is not contained in the outer 0 318 face, there must be an edge cd in D which intersects the partial segment of the edge ab

319 starting in a and ending in p, in its interior; see Figure 8b. Both of the edges bc, bd must 0 320 exist in the drawing D and cannot cross P . Consequently, by the properties of a topological

321 drawing, the triangle bcd (drawn blue) must contain a in its interior. Again, using the

322 properties of a topological drawing, we conclude that the edge ac in the original drawing D

323 (drawn red dashed) lies completely inside the triangle bcd: Due to adjacencies, the edge ac

324 cannot intersect the edges bc and cd. Consequently it also does not intersect bd. Now the

325 path P does not intersect ac, and the only edge of the triangle abc intersected by P is ab.

326 Therefore, x lies in the interior of the triangle abc. This contradicts the assumption that the

327 pair (D, x) violates the claim.