Sebastian Leipert

Advances in C-Planarity Testing of Clustered Graphs

A clustered graph C = (G, T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G = (V, E). Each vertex µ in T corresponds to a subset of the vertices of the graph called cluster. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automaticgraph drawing. The complexity status of c-planarity testing is unknown. It has been shown in [FCE95, Dah98] that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected.In this paper, we provide a polynomial time algorithm for c-planarity testing of almost c-connected clustered graphs, i.e., graphs for which all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each nonconnected cluster its super-cluster and all its siblings in T are connected. The algorithm is based on the concepts for the subgraph induced planar connectivity augmentation problem presented in [GJL+02]. We regard it as a first step towards general c-planarity testing.

Improving Walker's Algorithm to Run in Linear Time

The algorithm of Walker [5] is widely used for drawing trees of unbounded degree, and it is widely assumed to run in linear time, as the author claims in his article. But the presented algorithm clearly needs quadraticrun time. We explain the reasons for that and present a revised algorithm that creates the same layouts in linear time.

A new approach for visualizing UML class diagrams

UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and non-hierarchical relations poses a special challenge to a graph layout tool. Existing layout tools treat hierarchical and non-hierarchical relations either alike or as separate tasks in a two-phase process as in, e.g., [Seemann 1997]. We suggest a new approach for visualizing UML class diagrams leading to a balanced mixture of the following aesthetic criteria: Crossing minimization, bend minimization, uniform direction within each class hierarchy, no nesting of one class hierarchy within another, orthogonal layout, merging of multiple inheritance edges, and good edge labelling. We have realized our approach within the graph drawing library GoVisual. Experiments show the superiority to state-of-the-art and industrial standard layouts.

A characterization of level planar graphs

We present a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level non-planar (MLNP) subgraph patterns. We show that an MLNP subgraph pattern is completely characterized by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations.

A Fast Layout Algorithm for k-Level Graphs

We present a fast layout algorithm for k-level graphs with given permutations of the vertices on each level. The algorithm can be used in particular as a third phase of the Sugiyama algorithm [8]. In the generated layouts, every edge has at most two bends and is drawn vertically between these bends. The total length of short edges is minimized levelwise.

A note on computing a maximal planar subgraph using PQ-trees

The problem of computing a maximal planar subgraph of a nonplanar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQ-trees. The latest attempt has been reported by Jayakumar et al. In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and we note that the same holds for a modified version of the algorithm presented by Kant. Our conclusions most likely suggest not to use PQ-trees at all for this specific problem.

Automatic layout of UML class diagrams in orthogonal style

Unified modelling language (UML) diagrams have become increasingly important in engineering and re-engineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display generalizations, associations, aggregations, and compositions in one picture. The combination of directed and undirected relations poses a special challenge to a graph layout tool. Current approaches for the automatic layout of class diagrams are based on the layered graph drawing paradigm. These algorithms produce good results for class diagrams with large and deep structural information, that is, diagrams with a large and deep inheritance hierarchy. However, they do not perform satisfactorily in absence of this information. We suggest to use the topology-shape—metrics paradigm for automatic layout of class diagrams, which has been used very successfully for drawing undirected graphs in orthogonal style. Moreover, we introduce the algorithms UML-Kandinsky and GoVisual fitting into this paradigm. Both algorithms work for class diagrams with rich structural information as well as for class diagrams with few or no structural information. Therefore, they improve the existing algorithms significantly.

Graph Drawing Algorithm Engineering with AGD

We discuss the algorithm engineering aspects of AGD, a software library of algorithms for graph drawing. AGD represents algorithms as classes that provide one or more methods for calling the algorithm. There is a common base class, also called the type of an algorithm, for algorithms providing basically the same functionality. This enables us to exchange components and experiment with various algorithms and implementations of the same type. We give examples for algorithm engineering with AGD for drawing general non-hierarchical graphs and hierarchical graphs.

Subgraph Induced Planar Connectivity Augmentation

Given a planar graph G =( V, E) and a vertex set W ⊆ V , the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such that G =( V, E∪F ) is planar and the subgraph of G induced by W is connected. The problem arises in automatic graph drawing in the context of c-planarity testing of clustered graphs. We describe a linear time algorithm based on SPQR-trees that tests if a subgraph induced planar connectivity augmentation exists and, if so, constructs a minimum cardinality augmenting edge set.

Characterization of Level Non-planar Graphs by Minimal Patterns

In this paper we give a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level non-planar subgraph patterns (MLNP). We show that a MLNP is completely characterized by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations. These characterizations are an important first step towards attacking the NP-hard level planarization problem.