An annotated bibliography on 1-planarity

  title={An annotated bibliography on 1-planarity},
  author={Stephen G. Kobourov and Giuseppe Liotta and Fabrizio Montecchiani},

Beyond-Planarity: Density Results for Bipartite Graphs

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

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

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

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

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

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

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.

Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages

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.



On Drawings and Decompositions of 1-Planar Graphs

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

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

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.

A note on 1-planar graphs

NIC-planar graphs

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

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)

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.

The structure of 1-planar graphs

1-Planar Graphs have Constant Book Thickness

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

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.