Christian Bachmaier

A Radial Adaptation of the Sugiyama Framework for Visualizing Hierarchical Information

In radial drawings of hierarchical graphs, the vertices are placed on concentric circles rather than on horizontal lines and the edges are drawn as outward monotone segments of spirals rather than straight lines as it is done in the standard Sugiyama framework. This drawing style is well suited for the visualization of centrality in social networks and similar concepts. Radial drawings also allow a more flexible edge routing than horizontal drawings, as edges can be routed around the center in two directions. In experimental results, this reduces the number of crossings by approximately 30 percent on average. Few crossings are one of the major criteria for human readability. This paper is a detailed description of a complete framework for visualizing hierarchical information in a new radial fashion. Particularly, we briefly cover extensions of the level assignment step to benefit from the increasing perimeters of the circles, present three heuristics for crossing reduction in radial level drawings, and also show how to visualize the results

Gravisto: graph visualization toolkit

Gravisto, the Graph Visualization Toolkit, is more than a (Java-based) editor for graphs. It includes data structures, graph algorithms, several layout algorithms, and a graph viewer component. As a general toolkit for the visualization and automatic layout of graphs it is extensible with plug-ins and is suited for the integration in other Java-based applications.

Clustered Level Planarity

Planarity is an important concept in graph drawing. It is generally accepted that planar drawings are well understandable. Recently, several variations of planarity have been studied for advanced graph concepts such as k-level graphs [16, 15, 13, 14, 11, 12, 10, 6] and clustered graphs [7, 5]. In k-level graphs, the vertices are partitioned into k levels and the vertices of one level are drawn on a horizontal line. In clustered graphs, there is a recursive clustering of the vertices according to a given nesting relation. In this paper we combine the concepts of level planarity and clustering and introduce clustered k-level graphs. For connected clustered level graphs we show that clustered k-level planarity can be tested in \(\mathcal O(k|v|)\) time.

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

Drawing phylogenetic trees

\mathcal{NP}

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

Classification of planar upward embedding

We present linear-time algorithms for drawing phylogenetic trees in radial and circular representations. In radial drawings given edge lengths (representing evolutionary distances) are preserved, but labels (names of taxons represented in the leaves) need to be adjusted, whereas in circular drawings labels are perfectly spread out, but edge lengths adjusted. Our algorithms produce drawings that are unique solutions to reasonable criteria and assign to each subtree a wedge of its own. The linear running time is particularly interesting in the circular case, because our approach is a special case of Tuttes barycentric layout algorithm involving the solution of a system of linear equations.

Cyclic Leveling of Directed 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.

Drawing Recurrent Hierarchies

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.

A global k -level crossing reduction algorithm

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