Embedding Graphs into Embedded Graphs

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle

Constrained Planarity in Practice -- Engineering the Synchronized Planarity Algorithm

In the constrained planarity setting, we ask whether a graph admits a planar drawing that additionally satisfies a given set of constraints. These constraints are often derived from very natural problems; prominent examples are Level Planarity, where vertices have to lie on given horizontal lines indicating a hierarchy, and Clustered Planarity, where we additionally draw the boundaries of clusters which recursively group the vertices in a crossing-free manner. Despite receiving significant amount of attention and substantial theoretical progress on these problems, only very few of the found solutions have been put into practice and evaluated experimentally. In this paper, we describe our implementation of the recent quadratic-time algorithm by Bl\"asius et al. [TALG Vol 19, No 4] for solving the problem Synchronized Planarity, which can be seen as a common generalization of several constrained planarity problems, including the aforementioned ones. Our experimental evaluation on an existing benchmark set shows that even our baseline implementation outperforms all competitors by at least an order of magnitude. We systematically investigate the degrees of freedom in the implementation of the Synchronized Planarity algorithm for larger instances and propose several modifications that further improve the performance. Altogether, this allows us to solve instances with up to 100 vertices in milliseconds and instances with up to 100 000 vertices within a few minutes.Comment: to appear in Proceedings of ALENEX 202

Real-time interactive visualization of large networks on a tiled display system

This paper introduces a methodology for visualizing large real-world (social) network data on a high-resolution tiled display system. Advances in network drawing algorithms enabled real-time visualization and interactive exploration of large real-world networks. However, visualization on a typical desktop monitor remains challenging due to the limited amount of screen space and ever increasing size of real-world datasets.To solve this problem, we propose an integrated approach that employs state-of-the-art network visual-ization algorithms on a tiled display system consisting of multiple screens. Key to our approach is to use the machine's graphics processing units (GPUs) to their fullest extent, in order to ensure an interactive setting with real-time visualization. To realize this, we extended a recent GPU-based implementation of a force-directed graph layout algorithm to multiple GPUs and combined this with a distributed rendering approach in which each graphics card in the tiled display system renders precisely the part of the network to be displayed on the monitors attached to it.Our evaluation of the approach on a 12-screen 25 megapixels tiled display system with three GPUs, demonstrates interactive performance at 60 frames per second for real-world networks with tens of thousands of nodes and edges. This constitutes a performance improvement of approximately 4 times over a single GPU implementation. All the software developed to implement our tiled visualization approach, including the multi-GPU network layout, rendering, display and interaction components, are made available as open-source software.Computer Systems, Imagery and Medi