Wolfgang Brunner

Linear Time Planarity Testing and Embedding of Strongly Connected Cyclic Level Graphs

A level graph is a directed acyclic graph with a level assignment for each node. Such graphs play a prominent role in graph drawing. They express strict dependencies and occur in many areas, e. g., in scheduling problems and program inheritance structures. In this paper we extend level graphs to cyclic level graphs. Such graphs occur as repeating processes in cyclic scheduling, visual data mining, life sciences, and VLSI. We provide a complete study of strongly connected cyclic level graphs. In particular, we present a linear time algorithm for the planarity testing and embedding problem, and we characterize forbidden subgraphs. Our results generalize earlier work on level graphs.

Cyclic Leveling of Directed Graphs

The Sugiyama framework is the most commonly used concept for visualizing directed graphs. It draws them in a hierarchical way and operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles must be displayed as such, e. g., distinguished cycles in the biosciences and processes that repeat in a daily or weekly turn. This forbids the removal of cycles. In their seminal paper Sugiyama et al. also introduced recurrent hierarchies as a concept to draw graphs with cycles. However, this concept has not received much attention since then. In this paper we investigate the leveling problem for cyclic graphs. We show that minimizing the sum of the length of all edges is

Drawing Recurrent Hierarchies

{\mathcal{NP}}

A global k -level crossing reduction algorithm

-hard for a given number of levels and present three different heuristics for the leveling problem. This sharply contrasts the situation in the hierarchical style of drawing directed graphs, where this problem is solvable in polynomial time.

Coordinate Assignment for Cyclic Level Graphs

Directed graphs are generally drawn as level drawings using the hierarchical approach. Such drawings are constructed by a framework of algorithms which operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles should be displayed as such, e.g., distinguished cycles in the biosciences and scheduling processes repeating in a daily or weekly turn. In their seminal paper on hierarchical drawings Sugiyama et al. [31] also introduced recurrent hierarchies. This concept supports the drawing of cycles and their unidirectional display. However, it had not been investigated. In this paper we complete the cyclic approach and investigate the coordinate assignment phase. The leveling and the crossing reduction for recurrent hierarchies have been studied in the companion papers [3,4]. We provide an algorithm which runs in linear time and constructs an intermediate drawing with at most two bends per edge and aligned edge segments in an area of quadratic width times the preset number of levels height. This area bound is optimal for such drawings. Our approach needs new techniques for solving cyclic dependencies, such as skewing edges and cutting components. The drawings can be transformed into 2D drawings displaying all cycles counterclockwise around a center and into 3D drawings winding the cycles around a cylinder.

Plane drawings of queue and deque graphs

Directed graphs are commonly drawn by the Sugiyama algorithm, where crossing reduction is a crucial phase. It is done by repeated one-sided 2-level crossing minimizations, which are still

Cyclic level planarity testing and embedding

{\mathcal{NP}}

Tree Drawings on the Hexagonal Grid

-hard. We introduce a global crossing reduction, which at any particular time captures all crossings, especially for long edges. Our approach is based on the sifting technique and improves the level-by-level heuristics in the hierarchic framework by a further reduction of the number of crossings by 5 – 10%. In addition it avoids type 2 conflicts which help to straighten the edges, and has a running time which is quadratic in the size of the input graph independently of dummy vertices. Finally, the approach can directly be extended to cyclic, radial, and clustered level graphs where it achieves similar improvements over the previous algorithms.

Global k-Level Crossing Reduction

The Sugiyama framework is the most commonly used concept for visualizing directed graphs. It draws them in a hierarchical way and operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles must be displayed as such, e. g., distinguished cycles in the biosciences and scheduling processes which repeat in a daily or weekly turn. This excludes the removal of cycles. In their seminal paper Sugiyama et al. introduced recurrent hierarchies as a concept to draw graphs with cycles. However, this concept has not received much attention in the following years. In this paper we supplement our cyclic Sugiyama framework and investigate the coordinate assignment phase. We provide an algorithm which runs in linear time and constructs drawings which have at most two bends per edge and use quadratic area.

Grid sifting: Leveling and crossing reduction

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.