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The Density Formula: One Lemma to Bound Them All

Michael Kaufmann, Boris Klemz, Kristin Knorr, Meghana M. Reddy, Felix Schröder, Torsten Ueckerdt

Abstract

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called $k$-planar graphs with $k=1,2$, fan-crossing / fan-planar graphs, $k$-bend RAC-graphs with $k=0,1,2$, quasiplanar graphs, and $k^+$-real face graphs. In some cases ($1$-bend and $2$-bend RAC-graphs and fan-crossing / fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.

The Density Formula: One Lemma to Bound Them All

Abstract

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called -planar graphs with , fan-crossing / fan-planar graphs, -bend RAC-graphs with , quasiplanar graphs, and -real face graphs. In some cases (-bend and -bend RAC-graphs and fan-crossing / fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.
Paper Structure (20 sections, 29 theorems, 8 equations, 5 figures, 1 table)

This paper contains 20 sections, 29 theorems, 8 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. $k$-Bend RAC-Drawings.
  4. Fan-Crossing Drawings.
  5. Quasiplanar Drawings.
  6. Further applications.
  7. Terminology, Conventions, and Notation
  8. The Density Formula
  9. k-Bend RAC-Graphs
  10. Fan-Crossing Graphs
  11. Flaws in the Original Proofs from Related Work
  12. Quasiplanar Graphs
  13. Concluding Remarks
  14. Appendix
  15. k-Bend RAC-Graphs
  16. ...and 5 more sections

Key Result

Lemma 3

Let $t$ be a real number. Let $\Gamma$ be a connected drawing of a graph $G = (V,E)$ with at least one edge. Then

Figures (5)

  • Figure 1: Simple fan-planar drawings have neither configuration I, nor II, nor III. Simple fan-crossing drawings have no configuration I and no triangle-crossings.
  • Figure 2: Lenses with no vertex and no crossing in their interior. Such configurations are forbidden in non-homotopic drawings.
  • Figure 3: Left: Cells (and their sizes) that do not appear in simple or non-homotopic drawings on at least three vertices. Right: All types of cells $c$ of size $\|c\| \leq 5$ in a non-homotopic connected drawing on at least three vertices (cf. \ref{['obs:types-of-cells']}). The bottom row shows degenerate -cells, -cells, and -cells.
  • Figure 4: (Illustration of) a simple $2$-bend RAC-drawing of $G_4$ from \ref{['thm:2-bend-RAC-LB']}.
  • Figure 7: Left: Illustration of KaufUeck22 taken from the paper. Right: A counterexample.

Theorems & Definitions (29)

  • Lemma 3: Density Formula
  • Corollary 4
  • Corollary 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • Theorem 11
  • Lemma 12
  • ...and 19 more