Christopher Auer

1-Planarity of Graphs with a Rotation System

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. 1-planarity is known NP-hard, even for graphs of bounded bandwidth, pathwidth, or treewidth, and for near-planar graphs in which an edge is added to a planar graph. On the other hand, there is a linear time 1-planarity testing algorithm for maximal 1-planar graphs with a given rotation system. In this work, we show that 1-planarity remains NP-hard even for 3-connected graphs with (or without) a rotation system. Moreover, the crossing number problem remains NP-hard for 3-connected 1-planar graphs with (or without) a rotation system.

Recognizing Outer 1-Planar Graphs in Linear Time

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

Classification of planar upward embedding

\mathcal{NP}

On sparse maximal 2-planar graphs

-hard. Our main result is a linear-time algorithm that first tests whether a graphi¾?G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors see Fig. 3, which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p.

Plane drawings of queue and deque graphs

We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in ℝ3 and introduces new classes of planar upward drawable graphs, where some of them even allow cycles. Our approach leads to a classification of planar upward embeddability. In particular, we show the coincidence of the classes of planar upward drawable graphs on the sphere and on the standing cylinder. These classes coincide with the classes of planar upward drawable graphs with a homogeneous field on a cylinder and with a radial field in the plane. A cyclic field in the plane introduces the new class RUP of upward drawable graphs, which can be embedded on a rolling cylinder. We establish strict inclusions for planar upward drawability on the plane, the sphere, the rolling cylinder, and the torus, even for acyclic graphs. Finally, upward drawability remains NP-hard for the standing cylinder and the torus; for the cylinder this was left as an open problem by Limaye et al.

The duals of upward planar graphs on cylinders

A simple undirected graph G = (V, E) is k-planar if it can be drawn in the plane such that each edge is crossed at most k times, incident edges do not cross, and a pair of edges must not cross twice. Such graphs have attracted many graph drawers, see [1] and the references given there.

Upward planar graphs and their duals

In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue layout is a stack or queue graph, respectively. Typical stack and queue layouts are rainbows and twists visualizing the LIFO and FIFO principles, respectively. However, in such visualizations, twists cause many crossings, which make the drawings incomprehensible. We introduce linear cylindric layouts as a visualization technique for queue and deque (double-ended queue) graphs. It provides new insights into the characteristics of these fundamental data structures and extends to the visualization of mixed layouts with stacks and queues. Our main result states that a graph is a deque graph if and only if it has a plane linear cylindric drawing.

Rolling Upward Planarity Testing of Strongly Connected Graphs

We consider directed planar graphs with an upward planar drawing on the rolling and standing cylinders. These classes extend the upward planar graphs in the plane. Here, we address the dual graphs. Our main result is a combinatorial characterization of these sets of upward planar graphs. It basically shows that the roles of the standing and the rolling cylinders are interchanged for their duals.

Tight Upper Bounds for Minimum Feedback Arc Sets of Regular Graphs

We consider upward planar drawings of directed graphs in the plane (UP), and on standing (SUP) and rolling cylinders (RUP). In the plane and on the standing cylinder the edge curves are monotonically increasing in y-direction. On the rolling cylinder they wind unidirectionally around the cylinder. There is a strict hierarchy of classes of upward planar graphs: UP ? SUP ? RUP .In this paper, we show that rolling and standing cylinders switch roles when considering an upward planar graph and its dual. In particular, we prove that a strongly connected graph is RUP if and only if its dual is a SUP dipole. A dipole is an acyclic graph with a single source and a single sink. All RUP graphs are characterized in terms of their duals using generalized dipoles. Moreover, we obtain a characterization of the primals and duals of wSUP graphs which are upward planar graphs on the standing cylinder and allow for horizontal edge curves.

Optical graph recognition

A graph is upward planar if it can be drawn without edge crossings such that all edges point upward. Upward planar graphs have been studied on the plane, the standing and rolling cylinders. For all these surfaces, the respective decision problem \(\mathcal{NP}\)-hard in general. Efficient testing algorithms exist if the graph contains a single source and a single sink but only for the plane and standing cylinder.