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Saturated Drawings of Geometric Thickness k

Patricia Bachmann, Anna Brötzner, Miriam Goetze, Philipp Kindermann, Matthias Pfretzschner, Soeren Terziadis

TL;DR

This work studies saturated geometric drawings of graphs with geometric thickness $k$, focusing on convex and non-convex vertex placements. It introduces $\Theta^k$-drawings (with precoloring) and the notion of saturation, and derives both upper and lower bounds on edge density under these constraints. In the convex setting, it proves an overarching upper bound of $|E|\le n + k(n-3)$ and, for min-saturated convex precolored cases, a tighter bound of $\frac{1}{2}(k+4)(n-2)$; for $k=2$ it shows the exact density in the convex precolored case as $3n-6$, and for $k=3$ provides lower bounds $\frac{5}{2}n-6$ (precolored) and $\frac{7}{2}n-8$ (fully saturated). Extending to non-convex graphs with $k=2$, the paper establishes a sharp lower bound of $3n-6$ edges for saturated drawings, using an edge-extension and planarization argument. Together, these results advance understanding of saturation phenomena in geometric-thickness graph classes and raise open questions on tight convex bounds and hull-vertex configurations.

Abstract

We investigate saturated geometric drawings of graphs with geometric thickness $k$, where no edge can be added without increasing $k$. We establish lower and upper bounds on the number of edges in such drawings if the vertices lie in convex position. We also study the more restricted version where edges are precolored, and for $k=2$ the case for vertices in non-convex position.

Saturated Drawings of Geometric Thickness k

TL;DR

This work studies saturated geometric drawings of graphs with geometric thickness , focusing on convex and non-convex vertex placements. It introduces -drawings (with precoloring) and the notion of saturation, and derives both upper and lower bounds on edge density under these constraints. In the convex setting, it proves an overarching upper bound of and, for min-saturated convex precolored cases, a tighter bound of ; for it shows the exact density in the convex precolored case as , and for provides lower bounds (precolored) and (fully saturated). Extending to non-convex graphs with , the paper establishes a sharp lower bound of edges for saturated drawings, using an edge-extension and planarization argument. Together, these results advance understanding of saturation phenomena in geometric-thickness graph classes and raise open questions on tight convex bounds and hull-vertex configurations.

Abstract

We investigate saturated geometric drawings of graphs with geometric thickness , where no edge can be added without increasing . We establish lower and upper bounds on the number of edges in such drawings if the vertices lie in convex position. We also study the more restricted version where edges are precolored, and for the case for vertices in non-convex position.

Paper Structure

This paper contains 8 sections, 14 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Convex Drawings
  3. Bounds for Saturated Convex Theta-k-Drawings
  4. Edge-Density of Saturated Convex Theta-3-Drawings
  5. Moving towards non-convexity in the free setting for k=2
  6. Conclusion
  7. Full Proofs of Section 2
  8. Omitted Proofs of Section 3

Key Result

Proposition 2

Every convex $\Theta^k$-drawing of a graph $G$ on $n \geq 3$ vertices contains at most $n+k(n-3)$ edges.

Figures (5)

  • Figure 1: Taken from dillencourt2004geometric. A drawing of the non-planar graph $K_{6,6}$ witnessing $\bar{\theta}(K_{6,6}) = 2$.
  • Figure 2: (\ref{['fig:convex_k_min_upper_bound-1']}) A nice matching $M$ (green) and diagonals that could be added to a color class containing $M$ (dashed). (\ref{['fig:convex_k_min_upper_bound-2']}) The two nice matchings $L(M)$ (left) and $R(M)$ (right). (\ref{['fig:convex_k_min_upper_bound-3']}) A precolored $\Theta^3$-zigzag $\Gamma$. (\ref{['fig:convex_k_min_upper_bound-4']}) Recoloring yields a $\Theta^3$-drawing which contains $\Gamma$ and two more edges (dashed).
  • Figure 3: (\ref{['fig:thickness_3_convex_free-1']}) Five vertices on a face of size at least $5$ in $\Gamma_{\mathrm{r}}$ and their diagonals in $\Gamma$. The black edge is not present ins $\Gamma$. (\ref{['fig:thickness_3_convex_free-2']}) The corresponding vertex-coloring of the conflict graph $H$.
  • Figure 4: (\ref{['fig:nonConvex-1']}) A $\Theta^2$-drawing $\Gamma$ where the blue edges form an inner triangulation. (\ref{['fig:nonConvex-2']}) The corresponding drawing $\Lambda$. (\ref{['fig:nonConvex-3']}) Triangulating each inner cell of $\Lambda$ yields seven additional red edges. Note that, while the red edges do not form an inner triangulation of the whole graph, we now have at least $3n-6$ edges overall as desired.
  • Figure 5: Left: A drawing $\Lambda$ obtained after extending all inner red edges. Right: its corresponding planarization $H$. Note that the cells of $\Lambda$ correspond bijectively to the faces of the planarization.

Theorems & Definitions (22)

  • Proposition 2: bernhart1979book
  • Proposition 3
  • proof
  • Lemma 3: obs:face_complexity
  • theorem 4
  • corollary 1
  • Lemma 5
  • proof
  • theorem 6
  • theorem 7
  • ...and 12 more