Kathrin Hanauer

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 Outer 1-Planar Graphs in Linear Time

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

On sparse maximal 2-planar graphs

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

NIC-planar graphs☆

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

The duals of upward planar graphs on cylinders

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

Upward planar graphs and their duals

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

Rolling Upward Planarity Testing of Strongly Connected Graphs

edges and less than

Tight Upper Bounds for Minimum Feedback Arc Sets of Regular Graphs

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

Characterizing Planarity by the Splittable Deque

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.

Outer 1-Planar Graphs

A graph is outer 1-planar o1p if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1-planar graphs, whose recognition is