Franz J. Brandenburg

On the density of maximal 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity. Maximal 1-planar graphs have at most 4n−8 edges. We show that there are sparse maximal 1-planar graphs with only

Recognizing and Drawing IC-Planar Graphs

\frac{45}{17} n + \mathcal{O}(1)

Straight-Line Grid Drawings of 3-Connected 1-Planar Graphs

edges. With a fixed rotation system there are maximal 1-planar graphs with only

1-Planarity of Graphs with a Rotation System

\frac{7}{3} n + \mathcal{O}(1)

Recognizing Optimal 1-Planar Graphs in Linear Time

edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than

Recognizing Outer 1-Planar Graphs in Linear Time

\frac{21}{10} n - \mathcal{O}(1)

Graph clustering I: Cycles of cliques

edges and less than

The nearest neighbor Spearman footrule distance for bucket, interval, and partial orders

\frac{28}{13} n - \mathcal{O}(1)

On the relationship between k-planar and k-quasi planar graphs

edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.

NIC-planar graphs☆

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossedi¾?at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graphi¾?G withi¾?n vertices, we present an On-time algorithm that computes a straight-line drawing ofi¾?G in quadratic area, and an