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What induces plane structures in complete graph drawings?

Alexandra Weinberger, Ji Zeng

Abstract

This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.

What induces plane structures in complete graph drawings?

Abstract

This paper considers the task of connecting points on a piece of paper by drawing a curve between each pair of them. Under mild assumptions, we prove that many pairwise disjoint curves are unavoidable if either of the following rules is obeyed: any two adjacent curves do not cross, or any two non-adjacent curves cross at most once. Here, two curves are called adjacent if they share an endpoint. On the other hand, we demonstrate how to draw all curves such that any two adjacent curves cross exactly once, any two non-adjacent curves cross at least once and at most twice, and thus no two curves are disjoint. Furthermore, we analyze the emergence of disjoint curves without these mild assumptions, and characterize the plane structures in complete graph drawings guaranteed by each of the rules above.
Paper Structure (5 sections, 9 theorems, 1 equation, 5 figures)

This paper contains 5 sections, 9 theorems, 1 equation, 5 figures.

Table of Contents

  1. Introduction
  2. The adjacent-simple case
  3. The separate-simple case
  4. The main result
  5. Discussion

Key Result

Theorem 1

For any positive integer $m$, there exists $n$ such that every complete graph drawing on $n$ vertices that is either adjacent-simple or separate-simple must contain $m$ pairwise disjoint edges. On the other hand, there exist complete graph drawings on arbitrarily many vertices in which any two adjac

Figures (5)

  • Figure 1: Any two adjacent edges cross exactly once, any two non-adjacent edges cross at least once and at most twice.
  • Figure 2: A squid (left) and a caterpillar (right).
  • Figure 3: $R$ (marked with red stripes) is fully contained in $S$ (shaded in yellow).
  • Figure 4: A Type I drawing of vertices $v_1,v_2,\dots,v_6$ from left to right.
  • Figure 14: A loop labeled $v_iv_j$ is part of the edge between vertices $v_i$ and $v_j$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • Lemma 8
  • Proposition 9