Sabine Cornelsen

Visone Software for Visual Social Network Analysis

We are developing a social network tool that is powerful, comprehensive, and yet easy to use. The unique feature of our tool is the integration of network analysis and visualization. In a long-term interdisciplinary research collaboration, members of our group had implemented several prototypes to explore and demonstrate the feasibility of novel methods. These prototypes have been revised and combined into a stand-alone tool which will be extended regularly.

Track assignment

We consider a station in which several trains might stop at the same track at the same time. The trains might enter and leave the station from both sides, but the arrival and departure times and directions are fixed according to a given time table. The problem is to assign tracks to the trains such that they can enter and leave the station on time without being blocked by any other train. We consider some variation of the problem on linear time tables as well as on cyclic time tables and show how to solve them as a graph coloring problem on special graph classes. One of these classes are the so called circular arc containment graphs for which we give an optimal O(nlogn) coloring algorithm.

Drawing Graphs on Two and Three Lines

We give a linear-time algorithm to decide whether a graph has a planar LL-drawing, i.e. a planar drawing on two parallel lines. This has previously been known only for trees. We utilize this result to obtain planar drawings on three lines for a generalization of bipartite graphs, also in linear time.

Completely Connected Clustered Graphs

Planar drawings of clustered graphs are considered. We introduce the notion of completely connected clustered graphs, i.e. hierarchically clustered graphs that have the property that not only every cluster but also each complement of a cluster induces a connected subgraph. As a main result, we prove that a completely connected clustered graph is c-planar if and only if the underlying graph is planar. Further, we investigate the influence of the root of the inclusion tree to the choice of the outer face of the underlying graph and vice versa.

Path-based supports for hypergraphs

A path-based support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is NP-hard to decide whether a path-based support has a monotone drawing, to determine a path-based support with the minimum number of edges, or to decide whether there is a planar path-based support, we show that a path-based tree support can be computed in polynomial time if it exists.

More Canonical Ordering

Canonical ordering is an important tool in planar graph drawing and other applications. Although a linear-time algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly dened leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. Further, we discuss duality aspects and relations to Schnyder woods.

Blocks of hypergraphs: applied to hypergraphs and outerplanarity

A support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a connected subgraph. We show how to test in polynomial time whether a given hypergraph has a cactus support, i.e. a support that is a tree of edges and cycles. While it is NP-complete to decide whether a hypergraph has a 2-outerplanar support, we show how to test in polynomial time whether a hypergraph that is closed under intersections and differences has an outerplanar or a planar support. In all cases our algorithms yield a construction of the required support if it exists. The algorithms are based on a new definition of biconnected components in hypergraphs.

Phylogenetic graph models beyond trees

A graph model for a set S of splits of a set X consists of a graph and a map from X to the vertices of the graph such that the inclusion-minimal cuts of the graph represent S. Phylogenetic trees are graph models in which the graph is a tree. We show that the model can be generalized to a cactus (i.e. a tree of edges and cycles) without losing computational efficiency. A cactus can represent a quadratic rather than linear number of splits in linear space. We show how to decide in linear time in the size of a succinct representation of S whether a set of splits has a cactus model, and if so construct it within the same time bounds. As a byproduct, we show how to construct the subset of all compatible splits and a maximal compatible set of splits in linear time. Note that it is NP-complete to find a compatible subset of maximum size. Finally, we briefly discuss further generalizations of tree models.

Accelerated bend minimization

We present an

How to draw the minimum cuts of a planar graph

\mathcal O( n^{3/2})