Edge-Minimum Saturated k-Planar Drawings

@inproceedings{Chaplick2021EdgeMinimumSK,
  title={Edge-Minimum Saturated k-Planar Drawings},
  author={Steven Chaplick and Jonathan Rollin and Torsten Ueckerdt},
  booktitle={Graph Drawing},
  year={2021}
}
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… 
Saturated $2$-planar drawings with few edges
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
Saturated k-Plane Drawings with Few Edges
TLDR
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 35 REFERENCES
On topological graphs with at most four crossings per edge
Extending Simple Drawings
TLDR
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.
On the Density of Maximal 1-Planar Graphs
TLDR
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.
Improvements on the density of maximal 1‐planar graphs
TLDR
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.
Long cycles and spanning subgraphs of locally maximal 1‐planar graphs
TLDR
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.
A Survey of Minimum Saturated Graphs
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
Extending simple drawings with one edge is hard
TLDR
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.
The number of crossings in multigraphs with no empty lens
TLDR
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 \(\alpha >0\).
On the Maximum Number of Crossings in Star-Simple Drawings of Kn with No Empty Lens
TLDR
An upper bound of $3((n-4)!)$ is proved on the maximum number of crossings between any pair of edges in a star-simple drawing of a graph.
...
...