Beyond-planarity: A generalization of the planarity concept in graph drawing

超越平面性：图形绘制中平面性概念的概括

• 批准号: 364468267
• 负责人: Professor Dr. Michael Kaufmann, Ph.D.
• 金额: --
• 依托单位: Fachbereich IV: Informatik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/364468267?language=en
• 关键词: Beyond; planarity; generalization; concept; graph

The research field of graphs beyond planarity has been developed in recent years tremendously; evidence are several workshops and Dagstuhl seminars with this particular topic, as well as a survey book that will be published soon. In the first proposal in 2017, we gave a wide collection of research tasks in various directions. Meanwhile, we contributed in several directions; we summarized this in the progress report. However, we have also identified several new interesting research challenges and directions that we want to follow.1. Classification: We want robust definitions that are also parametrizable (fan-planar -> k-fan-planar, k-gap-planar -> (k,l)-gap-planar). We also want clear hierarchies between different graph classes. We expect to find new classes that complete the hierarchic structure. The classification will also be accompanied by combinatorial and algorithmic analyses on structural properties and parameters (e.g., low degree, girth etc) of the considered classes. 2. Layout: During the first project phase, we observed that the work on layout algorithms for graphs beyond planarity is very limited. The main reason for this is that the known techniques cannot be adopted so easily. Therefore, during the second project phase we will put our main focus on this aspect. For example for the standard approach to compute an appropriate topological embedding, and then a corresponding geometric embedding, both steps are challenging for the case of graphs beyond planarity and provide a considerable portion of risk in the project. To find algorithms which are practically applicable, we plan to provide fast exponential-time algorithms, maybe refined by parametrization, or efficient heuristics that can produce close-to-optimal layouts. It is definitely a far-from-trivial task to find corresponding requirements for more complex classes. Only elementary results are currently known mostly limited to the class of 1-planar graphs.3. Dissemination: By organizing meetings with other groups, we will develop the field further keeping the topic of beyond planarity as a central topic in regular workshops such as GNV in Heiligkreuztal, BWGD in Bertinoro and Dagstuhl. At such workshops, we find new insights by combining forces with people from combinatorial graph theory and computational geometry but also from algorithmic graph theory (e.g., Pach, T\'oth, Hoffmann, Speckmann). In 2016 and 2019, the applicant co-organized two Dagstuhl seminars on beyond planarity. A new edition of this successful Dagstuhl seminar is currently under consideration. As a side note, we further mention that in 2021 our group will be organizing the 29th Symposium of Graph Drawing and Network Visualization in Tübingen, a great honor and appreciation of our work in the field.

近年来，近年来已经开发了超出平面图的图形研究领域。证据是该特定主题的几个讲习班和dagstuhl精确的，以及将很快出版的调查书。在2017年的第一个提案中，我们在各个方向上提供了大量的研究任务。同时，我们在多个方向上做出了贡献。我们在进度报告中总结了这一点。但是，我们还确定了我们要遵循的几个新的有趣的研究挑战和方向。1。分类：我们需要也是可参数化的鲁棒定义（fan-planar-> k-fan-planar，k-gap-planar->（k，l）-GAP-Planar）。我们还希望在不同的图类别之间进行清晰的层次结构。我们希望找到完成层次结构的新类。该分类还将通过对所考虑类别的结构特性和参数（例如，低度，玫瑰等）进行组合和算法分析来完成。 2.布局：在第一个项目阶段，我们观察到，在平面度之外的图形上的布局算法上的工作非常有限。这样做的主要原因是，已知技术不能如此轻松地采用。因此，在第二个项目阶段，我们将主要关注这一方面。例如，对于计算适当的拓扑嵌入，然后是相应的几何嵌入的标准方法，对于超出平面性的图表而言，这两个步骤都构成了挑战，并在项目中提供了一部分风险。为了找到实际上适用的算法，我们计划提供快速的指数时间算法，这可能是通过参数或有效的启发式方法来完善的，这些算法可能会产生近距离的布局。找到更复杂类的相应要求绝对是一项遥不可及的任务。目前只有基本结果主要仅限于1平面图的类别3。传播：通过与其他团体组织会议，我们将在常规讲习班中，例如Heiligkreuztal的GNV，Bertinoro和Dagstuhl等常规讲习班中的“超越平面”作为中心主题。在这样的研​​讨会上，我们通过将力与组合图理论和计算几何形状的人结合起来，还通过算法图理论（例如Pach，T \'oth，Hoffmann，Speckmann）找到新的见解。在2016年和2019年，适用的共同组织了两个dagstuhl Semiars on Beyond Planarity。目前正在考虑这个成功的Dagstuhl Semiar的新版本。附带说明，我们进一步提到，在2021年，我们的小组将在Tübingen举办第29届图形图和网络可视化研讨会，这是对我们在该领域的工作的巨大荣誉和欣赏。