单调聚类层次平面性
Monotone Clustered Level Planarity
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中文总结 AI 辅助
研究层次平面性与聚类平面性组合的单调聚类层次平面性问题，证明其NP完全，通过分析不同参数化情况，表明除顶点覆盖结合聚类数外，多数图结构参数下无固定参数可处理性，同时证明以顶点覆盖数和聚类数参数化时具有固定参数可处理性。
中文摘要 AI 辅助
我们考虑层次平面性和聚类平面性这两个受限平面性问题的组合。传统上，在此背景下研究的是具有凸聚类的层次平面绘制。Fink等人（EuroCG 2024）最近引入了一种不同的层次平面性和聚类平面性组合方式，通过在层次平面设置中模仿聚类平面性的经典特征：问题（y -）单调聚类层次平面性（mCLP）寻求一种层次平面绘制，其中可以用不交叉聚类边界的边扩充每个聚类，使其在保持层次平面性的同时变得连通。Fink等人（EuroCG 2024）表明，即使对于双连通单源图以及具有恒定层数和聚类数的实例，mCLP也是NP完全的。我们进一步对mCLP问题的参数化复杂度进行分类，一方面表明即使对于由有界大小的树组成的森林、无孤立顶点且聚类数或层数为小常数的实例，该问题也具有难度。这排除了几乎所有图结构参数的固定参数可处理性，除了顶点覆盖，即使结合聚类数也是如此。我们通过表明以顶点覆盖数和聚类数进行参数化时具有固定参数可处理性来补充这一点。一个主要障碍是mCLP是非遗传的，即肯定实例的子实例可能是否定实例，反之亦然，这使得应用通常的归约技术具有挑战性。
英文摘要
We consider the combination of the two constrained planarity problems Level- and Clustered Planarity. Traditionally, level-planar drawings with convex clusters have been studied in this setting. Fink et al. (EuroCG 2024) recently introduced a different way of combining level- and clustered planarity by mimicking a classic characterization of clustered planarity in the level-planar setting: The problem (y-)monotone Clustered Level Planarity (mCLP) seeks a level-planar drawing in which it is possible to augment each cluster with edges that do not cross cluster boundaries so that it becomes connected while maintaining level-planarity. This is in line with previous research on clustered planarity that poses certain requirements on the augmentation edges that make each cluster connected, e.g., that they form a path. Fink et al. (EuroCG 2024) showed that mCLP is NP-complete even for biconnected single-source graphs and instances with a constant number of levels and clusters. We further classify the parameterized complexity of the mCLP problem by, on the one hand, showing hardness even for instances that consist of a forest with trees of bounded size, no isolated vertices, and a small constant number of either clusters or levels. This excludes fixed-parameter tractability for almost all graph-structural parameters, except for vertex cover, even in conjunction with the number of clusters. We complement this by showing fixed-parameter tractability when parameterizing by the vertex cover number and the number of clusters. A major obstacle is the fact that mCLP is non-hereditary, i.e., subinstances of yes-instances may be no-instances and vice versa, which makes it challenging to apply usual reduction techniques.
