Topological Drawings meet Classical Theorems from Convex Geometry

TL;DR

This work investigates how classical convex geometry theorems translate to simple topological drawings of the complete graph $K_n$ by leveraging triangles as convex-like cells and introducing generalized signotopes as a combinatorial encoding. It proves a purely combinatorial generalization of Kirchberger's theorem for generalized signotopes, connects these objects to topological drawings, and analyzes Carathéodory-type results, Colorful Carathéodory, and Helly-type phenomena within this convexity framework. The authors show that cell-convex drawings can exhibit arbitrarily large Helly numbers and that Colorful Carathéodory holds for pseudolinear drawings but not beyond, highlighting the limits of these theorems in broader drawing classes. They also develop a structural and enumerative theory for generalized signotopes, establishing tight asymptotic bounds $g(n)=2^{\Theta(n^3)}$, exploring flip-equivalence, and demonstrating that most generalized signotopes do not arise from topological drawings, with detailed small-$n$ classifications and constructions. These results deepen the understanding of how geometric- combinatorial representations encode planar graph drawings and open questions about drawable signotopes and broader extensions of convexity theorems in topological settings.

Abstract

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger's Theorem that is of purely combinatorial nature. It turned out that this classical theorem also applies to "generalized signotopes" - a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by the name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.

Figures (9)

• Figure 1: Forbidden patterns in topological drawings: self-crossings, double-crossings, touchings, and crossings of adjacent edges.
• Figure 2: Two weakly isomorphic drawings of $K_6$, which can be transformed into each other by a triangle-flip.
• Figure 3: The three types of topological drawings of $K_4$ in the plane.
• Figure 4: \ref{['fig:kirchberger_reverse_fig']} Simple drawing showing that the reverse direction of Kirchberger is not true. The bold edge is a separator for the drawing on all 6 vertices. However, the subdrawing of the $K_4$ marked with the dashed edges has no separator. The vertices of $A$ are marked red and the vertices of $B$ blue. \ref{['fig:kirchberger_reverse_orientation']} Orientations of the drawing yielding the generalized signotope $\gamma$.
• Figure 5: \ref{['fig:Caratheodory_illustration_page1']} and \ref{['fig:Caratheodory_illustration_page2']} give an illustration of the proof of Theorem \ref{['theorem:Caratheodory_generalized']}.

Theorems & Definitions (17)

• Proposition 1
• Theorem 1: Kirchberger for Generalized Signotopes
• proof : Proof of Theorem \ref{['theorem:kirchberger']}
• Theorem 2: Carathéodory for Topological Drawings BalkoFulekKyncl2015
• proof : Proof of Theorem \ref{['theorem:Caratheodory_generalized']}
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 3