# Edge-Minimum Saturated k-Planar Drawings

@inproceedings{Chaplick2020EdgeMinimumSK,
title={Edge-Minimum Saturated k-Planar Drawings},
author={Steven Chaplick and Jonathan Rollin and Torsten Ueckerdt},
booktitle={International Symposium Graph Drawing and Network Visualization},
year={2020}
}
• Published in
International Symposium Graph…
15 December 2020
• Mathematics
For a class $\mathcal{D}$ of drawings of loopless multigraphs in the plane, a drawing $D \in \mathcal{D}$ is saturated when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$. This is analogous to saturated graphs in a graph class as introduced by Turan (1941) and Erdős, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $\mathcal{D}$ of all $k$-planar drawings obeying a…
2 Citations
• Mathematics
• 2021
A drawing of a graph is k-plane if every edge contains at most k crossings. A k-plane drawing is saturated if we cannot add any edge so that the drawing remains k-plane. It is well-known that
• Mathematics
ArXiv
• 2020
This paper studies saturated $k-plane drawings with few edges, in which no edge can be added without violating$k$-planarity, and presents constructions withFew edges for different values of$ k$and$\ell$. ## References SHOWING 1-10 OF 34 REFERENCES • Mathematics Graph Drawing • 2019 It is proved that deciding if a given set of edges can be inserted into a simple drawing is NP-complete and that the maximization version of the problem is APX-hard. • Mathematics GD • 2012 It is shown that there are sparse maximal 1-planar graphs with only$\frac{45}{17} n + \mathcal{O}(1)$edges, and it is proved that a maximal 1 -planar rotation system of a graph uniquely determines its 1- Planar embedding. • Mathematics J. Graph Theory • 2018 A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once and a maximal 1-plane graph has at least$2.1n-O(1)$edges. • Mathematics J. Graph Theory • 2020 This work shows the existence of a spanning$3-connected planar subgraph and proves that G is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that the graph is traceable if G has at least four four $3#-ver Tex-cuts. • Mathematics • 2011 Given a family of (hyper)graphs$\mathcal{F}$a (hyper)graph$G$is said to be$\mathcal{F}$-saturated if$G$is$F$-free for any$F \in\mathcal{F}$but for any edge e in the complement of$G\$ the
• Mathematics
ArXiv
• 2019
It is NP-complete to decide whether a given edge can be inserted into a simple drawing, by this solving an open question by Arroyo, Derka, and Parada.
• Mathematics
GD
• 2018
Pach and Toth extended the Crossing Lemma of Ajtai et al. by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in G is at least $$\alpha e^3/n^2$$, for a suitable constant $$\alpha >0$$.
• Mathematics
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A long-standing problem about the maximal number of edges of a graph not containing a cycle of length 4 is solved and some unsolved problems are mentioned.