3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Straightening out planar poly-line drawings

We show that any $y$-monotone poly-line drawing can be straightened out while maintaining $y$-coordinates and height. The width may increase much, but we also show that on some graphs exponential width is required if we do not want to increase the height. Likewise $y$-monotonicity is required: there are poly-line drawings (not $y$-monotone) that cannot be straightened out while maintaining the height. We give some applications of our result.Comment: The main result turns out to be known (Pach & Toth, J. Graph Theory 2004, http://onlinelibrary.wiley.com/doi/10.1002/jgt.10168/pdf

05191 Abstracts Collection -- Graph Drawing

From 08.05.05 to 13.05.05, the Dagstuhl Seminar 05191 ``Graph Drawing\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Colored anchored visibility representations in 2D and 3D space

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In a visibility representation of a graph G, the vertices are represented by nonoverlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G, the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given.Peer ReviewedPostprint (author's final draft

On Optimal 2- and 3-Planar Graphs

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and T\'oth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple $2$- and $3$-planar graphs without homotopic parallel edges and self-loops. In this paper, we completely characterize optimal $2$- and $3$-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes

Visualization of graphs and trees for software analysis

A software architecture is an abstraction of a software system, which is indispensable for many software engineering tasks. Unfortunately, in many cases information pertaining to the software architecture is not available, outdated, or inappropriate for the task at hand. The RECONSTRUCTOR project focuses on software architecture reconstruction, i.e., obtaining architectural information from an existing system. Our research, which is part of RECONSTRUCTOR, focuses on interactive visualization and tries to answer the following question: How can users be enabled to understand the large amounts of information relevant for program understanding using visual representations? To answer this question, we have iteratively developed a number of techniques for visualizing software systems. A large number of these cases consists of hierarchically organized data, combined with adjacency relations. Examples are function calls within a hierarchically organized software system and correspondence relations between two different versions of a hierarchically organized software system. Hierarchical Edge Bundles (HEBs) are used to visualize adjacency relations in hierarchically organized data, such as the aforementioned function calls within a software system. HEBs significantly reduce visual clutter by visually bundling relations together. Massive Sequence Views (MSVs) are used in conjunction with HEBs to enable analysis of sequences of relations, such as function-call traces. HEBs are furthermore used to visually compare hierarchically organized data, e.g., two different versions of a software system. HEBs visually emphasize splits, joins, and relocations of subhierarchies and provide for interactive selection of sets of relations. Since HEBs require a hierarchy to perform the bundling, we present Force-Directed Edge Bundles (FDEBs) as an alternative to visually bundle relations together in the absence of a hierarchical component. FDEBs use a self-organizing approach to bundling in which edges are modeled as flexible springs that can attract each other. As a result, visual clutter is reduced and high-level edge patterns are better visible. Finally, in all these methods, a clear depiction of the direction of edges is important. We have therefore performed a separate study in which we evaluated ten representations (including the standard arrow) for depicting directed edges in a controlled user study