## 105 Citations

### Beyond-Planarity: Density Results for Bipartite Graphs

- MathematicsArXiv
- 2017

Borders on the number of edges that are tight up to small additive constants are proved for bipartite topological graphs; some of them are surprising and not along the lines of the known results for non-bipartite graphs.

### 1-Planar RAC Drawings with Bends

- Mathematics
- 2018

This thesis concerns the relationships among beyond-planar graphs, which generalize the planar graphs. In particular, it is about RAC drawings of 1-planar graphs and NIC-planar graphs in bounded area…

### Efficiently Partitioning the Edges of a 1-Planar Graph into a Planar Graph and a Forest

- Mathematics, Computer ScienceISAAC
- 2021

This paper reprove Ackerman’s result and shows that the split can be found in linear time by using an edge-contraction data structure by Holm, Italiano, Karczmarz, Łącki, Rotenberg and Sankowski.

### An Experimental Study of a 1-planarity Testing and Embedding Algorithm

- Computer ScienceWALCOM
- 2020

This work investigates the feasibility of a $1-planarity testing and embedding algorithm based on a backtracking strategy and shows that it can be successfully applied to graphs with up to 30 vertices, but suggests the need of more sophisticated techniques to attack larger graphs.

### On Optimal Beyond-Planar Graphs

- Mathematics
- 2022

The range for optimal graphs is computed, combinatorial properties are established, and it is shown that every graph is a topological minor of an optimal graph.

### A Structure of 1-Planar Graph and Its Applications to Coloring Problems

- MathematicsGraphs Comb.
- 2019

It is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most 2p-2, and it is shown thatevery 1- Planar graph has an equitable edge coloring with k colors for any integer k.

### Graph Planarity by Replacing Cliques with Paths

- MathematicsAlgorithms
- 2020

It is proved that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G isA simple 1-planegraph, for any value of h.

### On 3D visibility representations of graphs with few crossings per edge

- MathematicsTheor. Comput. Sci.
- 2019

### Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages

- MathematicsSoCG
- 2020

It is proved that this family of graphs has bounded book thickness, and as a corollary, the first constant upper bound for the book thickness of optimal 2-planar graphs is obtained.

## References

SHOWING 1-10 OF 185 REFERENCES

### On Drawings and Decompositions of 1-Planar Graphs

- MathematicsElectron. J. Comb.
- 2013

It is demonstrated that each subgraph of an optimal 1-planar graph can be decomposed into a planar graph and a forest, and an upper bound on the number of edges of bipartite 1- Planar graphs is derived.

### 1-Planarity of Graphs with a Rotation System

- MathematicsJ. Graph Algorithms Appl.
- 2015

It is shown that 1-planarity remains NP-hard even for 3-connected 2-planar graphs with (or without) a rotation system, and the crossing number problem remainsNP-hard for3-connected 1- PLANAR graphs with a given rotation system.

### Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard

- MathematicsSIAM J. Comput.
- 2013

A new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs is obtained, and the concept of anchored embedding is introduced.

### On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

- MathematicsDiscuss. Math. Graph Theory
- 2016

It is proved that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2n rather than 3n, where n denotes the order of a graph.

### Testing Maximal 1-Planarity of Graphs with a Rotation System in Linear Time - (Extended Abstract)

- MathematicsGraph Drawing
- 2012

The problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system (i.e., the circular ordering of edges for each vertex) is given.

### 1-Planar Graphs have Constant Book Thickness

- MathematicsESA
- 2015

It is proved that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages.

### Drawing Outer 1-planar Graphs with Few Slopes

- MathematicsJ. Graph Algorithms Appl.
- 2015

It is shown that an outer 1-planar graph G of bounded degree Δ admits an outer 3-line drawing that uses OΔ different slopes, which extends a previous result by Knauer et al. about the planar slope number of outerplanar graphs CGTA.