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
- Mathematics, Computer ScienceISAAC
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.
1-planarity testing and embedding: An experimental study
- Computer ScienceComput. Geom.
An Experimental Study of a 1-planarity Testing and Embedding Algorithm
- Computer ScienceWALCOM
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
- MathematicsGraphs Comb.
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.
On 3D visibility representations of graphs with few crossings per edge
- MathematicsTheor. Comput. Sci.
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.
SHOWING 1-10 OF 185 REFERENCES
On Drawings and Decompositions of 1-Planar Graphs
- MathematicsElectron. J. Comb.
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.
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.
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
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
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
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.
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.