Andreas Gleißner

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

COMPARING AND AGGREGATING PARTIAL ORDERS WITH KENDALL TAU DISTANCES

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

1-Planarity of Graphs with a Rotation System

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

Classification of planar upward embedding

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

On sparse maximal 2-planar graphs

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

Plane drawings of queue and deque graphs

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

The duals of upward planar graphs on cylinders

edges and less than

Comparing and aggregating partial orders with kendall tau distances

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

Upward planar graphs and their duals

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.

Grid sifting: Leveling and crossing reduction

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order or a bucket order of all candidates, where bucket orders allow ties. In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard. The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in time for two voters. We show that it is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in , but not in NP or coNP unless NP = coNP, even for four voters.