向上性的代价
The Price of Upwardness
AI总结：
本文研究DAG的向上k-平面绘制，证明多类图的边交叉数无界但外路径图至多为二，并证明k=1时测试向上k-平面性已NP完全，同时给出单源DAG的线性时间判定算法。
摘要：
并非每个底层无向图为平面图的有向无环图（DAG）都允许向上平面绘制。我们感兴趣的是通过考虑DAG的向上k-平面绘制，将向上绘制的概念推广到平面性之外，在这种绘制中，边沿共同方向单调递增，并且对于某个整数k≥1，每条边最多被交叉k次。我们证明，对于二分外平面、立方或有界路径宽度DAG类，单调绘制中每条边的交叉数通常是无界的。然而，对于外路径图，它至多为二，并且一般情况下它至多与带宽的平方成正比。从计算的角度来看，我们证明测试向上k-平面性对于k=1已经是NP完全的，甚至对于向上平面性测试是多项式的受限实例也是如此。在积极的方面，我们可以在线性时间内判定一个单源DAG是否允许所有顶点都与外部面关联的向上1-平面绘制。
英文摘要：
Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that testing upward-$k$-planarity is NP-complete already for $k=1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.
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