arXiv subjects
Publications and source records attributed to Franz J. Brandenburg.
We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal 1-planar, respectively, with a polynomial of degree 120, 3, and 5, respectively.
A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices; beyond that segments do not intersect. We generalize visibility to 1-visibility, where each edge- (vertex-) segment crosses at most one vertex- (edge-) segment. In other words, a vertex is crossed by at most one edge, and vice-versa. We show that 1-visibility properly extends 1-planarity and develop a linear time algorithm to compute a 1-visibility representation of an embedded 1-planar graph on O(n^2) area. A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. Concerning density, both 1-visible and 1-planar graphs of size $n$ have at most 4n-8 edges. However, for every n >= 7 there are 1-visible graphs with 4n-8 edge which are not 1-planar.
In order to evaluate, compare, and tune graph algorithms, experiments on well designed benchmark sets have to be performed. Together with the goal of reproducibility of experimental results, this creates a demand for a public archive to gather and store graph instances. Such an archive would ideally allow annotation of instances or sets of graphs with additional information like graph properties and references to the respective experiments and results. Here we examine the requirements, and introduce a new community project with the aim of producing an easily accessible library of graphs. Through successful community involvement, it is expected that the archive will contain a representative selection of both real-world and generated graph instances, covering significant application areas as well as interesting classes of graphs.