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Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach

Markus Chimani, Torben Donzelmann, Nick Kloster, Melissa Koch, Jan-Jakob Völlering, Mirko H. Wagner

Abstract

Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.

Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach

Abstract

Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.
Paper Structure (16 sections, 27 theorems, 3 equations, 5 figures, 1 table)

This paper contains 16 sections, 27 theorems, 3 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Framework for Proving Lower Bounds on Crossing Ratios
  4. Proving Crossing Ratios
  5. k-Planar
  6. k-Vertex-Planar
  7. IC-Planar
  8. NIC-Planar
  9. NNIC-Planar and k-Fan-Crossing-Free
  10. Fan-Planar and Variants
  11. k-Edge-Crossing
  12. k-Gap-Planar
  13. k-Apex-Planar and Skewness-k
  14. Rectilinear Crossing Ratios
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2

For every $\ell \geq 41$, there exists a $k$-planar graph $G_{\ell}$ with $n\in \Theta(\ell^2 k)$ vertices such that $\mathrm{cr}_{k\mathrm{\text{-}pl}\xspace}\xspace(G_{\ell}\xspace) \in \Omega(nk)$ and $\mathrm{cr}_{}\xspace(G_{\ell}\xspace) \in \mathcal{O}\xspace(k)$. Thus $\varrho_{k\mathrm{\tex

Figures (5)

  • Figure 1: Red-yellow (left) and blue-blue (right) drawing of $F$ used for standard drawings of $G_{\ell}\xspace$.
  • Figure 2: Visualizations of standard drawings of crossing con-graphs, for different beyond planarity concepts. The con-graphs are colored consistently to their color type (blue, gray, red, yellow); we use two shades of blue when showing crossing blue con-graphs. The depicted values for $\ell$ are chosen for easy comprehension of the drawing paradigm (e.g., (a) shows $\ell=3$, $k=2$); they are smaller than required in the proofs.
  • Figure 3: A fan-crossing-free but not NNIC-planar graph. The red edges cannot be crossed in any FCF drawing. The depicted fan-crossing-free subembedding of the non-gray edges is unique (up to mirroring), but not NNIC-planar: arrows point to two crossings that share three (shaded) vertices.
  • Figure 4: Drawings of the differently colored frame used in \ref{['thm:kgap:lb', 'thm:kapex:lb', 'thm:skewk:lb']} for the $k$-gap-planar, $k$-apex-planar, and skewness-$k$ crossing ratios, respectively.
  • Figure 5: Labeled version of \ref{['fig:nnic_vs_fcf']}: A fan-crossing free but not NNIC-planar graph $G=(V_L\cup V_S, E_W\cup E_G\cup E_L)$. Large vertices $V_L$ are drawn as large circles, small vertices $V_L$ as small gray dots. Wall edges $E_W$ are drawn red; guards $E_G$ gray; loners $E_L$ violet. Once it is established that walls can never be crossed, the labels at the loners gives the order used to argue that the fan-crossing-free embedding is unique (up to mirroring and irrelevant details for the guards). The embedding is not NNIC-planar: arrows point to two crossings that share three vertices (shaded in light violet).

Theorems & Definitions (27)

  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • Theorem 11
  • ...and 17 more