标题： 绘制Reeb图 显示英文标题

标题： Drawing Reeb Graphs

摘要： 叶状图是一种简单的拓扑描述符，在诸如拓扑数据分析和计算几何等诸多领域都有应用。 尽管叶状图很常见，但对其可视化却关注较少。 本文通过研究绘制叶状图的复杂性，在文献中填补了一个重要的空白。 具体来说，我们证明了对于直线边和曲线边，最小化叶状图交叉数的问题是NP难的。 另一方面，我们确定了一些特定类别的叶状图，即路径和毛虫图，它们可以实现无交叉的绘制。 我们还给出了一个最优算法，用于以最少的交叉数绘制环形叶状图，并对多环叶状图的绘制复杂性进行了初步观察。 我们希望这项工作能奠定理解叶状图可视化中固有图绘制挑战的基础，并为该领域的未来研究铺平道路。 显示英文摘要

摘要： Reeb graphs are simple topological descriptors with applications in many areas like topological data analysis and computational geometry. Despite their prevalence, visualization of Reeb graphs has received less attention. In this paper, we bridge an essential gap in the literature by exploring the complexity of drawing Reeb graphs. Specifically, we demonstrate that Reeb graph crossing number minimization is NP-hard, both for straight-lined and curved edges. On the other hand, we identify specific classes of Reeb graphs, namely paths and caterpillars, for which crossing-free drawings exist. We also give an optimal algorithm for drawing cycle-shaped Reeb graphs with the least number of crossings and provide initial observations on the complexities of drawing multi-cycle Reeb graphs. We hope that this work establishes the foundation for an understanding of the graph drawing challenges inherent in Reeb graph visualization and paves the way for future work in this area.