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Arrangements of Pseudocircles: On Digons and Triangles

Stefan Felsner, Sandro Roch, Manfred Scheucher

Abstract

In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972). First, we discuss the maximum number of digons or touching points. Grünbaum conjectured that there are at most $2n - 2$ digon cells or equivalently at most $2n - 2$ touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of $2n - 2$ touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Grünbaum conjectured that such arrangements have $2n - 4$ triangles. Snoeyink and Hershberger (1991) established a lower bound of $\lceil \frac{4}{3}n \rceil$. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only $\lceil \frac{16}{11}n \rceil$ triangles. We provide a construction which shows that $\lceil \frac{4}{3}n \rceil$ is the correct value.

Arrangements of Pseudocircles: On Digons and Triangles

Abstract

In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972). First, we discuss the maximum number of digons or touching points. Grünbaum conjectured that there are at most digon cells or equivalently at most touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Grünbaum conjectured that such arrangements have triangles. Snoeyink and Hershberger (1991) established a lower bound of . Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only triangles. We provide a construction which shows that is the correct value.
Paper Structure (9 sections, 11 theorems, 5 equations, 18 figures)

This paper contains 9 sections, 11 theorems, 5 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Digons and touchings
  3. Triangles in digon-free arrangements
  4. Related Work and Discussion
  5. Proof of Theorem \ref{['thm:digons']}
  6. Proof of Proposition \ref{['prop:trianglefree_tight']}
  7. Digon-free arrangements with few triangles
  8. Proof of Theorem \ref{['thm:triangles']} and Theorem \ref{['thm:extension_to_counterexample']}
  9. Proof of Proposition \ref{['prop:nosubN6family']}

Key Result

Theorem 2

Let ${\cal A}$ be a simple arrangement of $n$ pairwise intersecting pseudocircles. If the touching graph ${T}({\cal A})$ contains a triangle, then there exist at most $2n-2$ touchings, i.e., $p_2(\mathcal{A}) \le 2n-2$.

Figures (18)

  • Figure 1: An arrangement of $n\ge 4$ pairwise intersecting pseudocircles with exactly $2n-2$ digons. Digons are highlighted gray (Example copied from Grünbaum Gruenbaum1972).
  • Figure 2: Contracting some digons to touchings.
  • Figure 3: \ref{['fig:K3_labels_a']} An illustration of the subarrangement ${\cal K}$. \ref{['fig:K3_labels_b']} and \ref{['fig:K3_labels_c']}, respectively, illustrate an additional pseudocircle $C$ (red), the pc-arcs inside $\triangle\cup\triangle'$ are highlighted.
  • Figure 4: \ref{['fig:K3_arc_type_proof_1']}--\ref{['fig:K3_arc_type_proof_4']} illustrate Cases 1--4 from the proof of Claim \ref{['claim:same_arc_type']}. The pseudocircles $C$ and $C'$ are highlighted blue and red, respectively. The pc-arcs $A$ and $A'$ are emphasized.
  • Figure 5: Original arrangement ${\cal A}$
  • ...and 13 more figures

Theorems & Definitions (19)

  • Conjecture 1: Grünbaum's digon conjecture Gruenbaum1972
  • Theorem 2
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Conjecture 5: Weak Grünbaum triangle conjecture
  • Proposition 5
  • Conjecture 6: FelsnerScheucher2020
  • Theorem 6
  • proof
  • ...and 9 more