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

近年来，超平面图的研究领域得到了极大的发展。证据是关于这一特定主题的若干讲习班和Dagstuhl研讨会，以及即将出版的一本概览书。在2017年的第一个提案中，我们在各个方向上都给出了广泛的研究任务。与此同时，我们在几个方面做出了贡献；我们在进度报告中总结了这一点。然而，我们也确定了几个新的有趣的研究挑战和我们想要遵循的方向。分类：我们需要可参数化的鲁棒定义（fan-planar -> k-fan-planar, k-gap-planar -> (k,l)-gap-planar）。我们还希望不同的图类之间有清晰的层次结构。我们期望找到新的类来完成这个层次结构。分类还将伴随着对所考虑的类别的结构特性和参数（例如，低度，周长等）的组合和算法分析。2. 布局：在项目的第一个阶段，我们观察到对于平面以外图形的布局算法的工作非常有限。其主要原因是，已知的技术不能那么容易地被采用。因此，在第二个项目阶段，我们将把重点放在这方面。例如，对于计算适当的拓扑嵌入的标准方法，然后是相应的几何嵌入，这两个步骤对于超出平面性的图来说都是具有挑战性的，并且在项目中提供了相当大的风险。为了找到实际适用的算法，我们计划提供快速的指数时间算法，可能通过参数化改进，或者可以产生接近最优布局的有效启发式算法。为更复杂的类找到相应的需求，这绝对是一项艰巨的任务。目前已知的基本结果大多局限于一类平面图。传播：通过与其他团体组织会议，我们将进一步发展这一领域，在海利克罗伊茨塔尔的GNV、伯蒂诺罗和达格斯图尔的BWGD等定期研讨会上，将超越平面性的主题作为中心主题。在这样的研讨会上，我们通过与来自组合图论和计算几何以及算法图论（例如，Pach, T\'oth, Hoffmann, Speckmann）的人结合力量，发现了新的见解。在2016年和2019年，申请人共同组织了两次关于超越平面性的Dagstuhl研讨会。目前正在考虑为这次成功的达格施图尔讨论会再版。作为旁注，我们进一步提到，在2021年，我们的小组将在<s:1>宾根组织第29届图形绘制和网络可视化研讨会，这是对我们在该领域工作的巨大荣誉和赞赏。

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.