图的向上三维网格图
Upward Three-Dimensional Grid Drawings of Graphs
摘要：
图的\emph{three-dimensional grid drawing}是将顶点放置在具有整数坐标的不同点处，使得表示边的直线段不交叉。我们的目标是生成具有小包围盒体积的三维网格图。证明了每一个有界退化的$n$ -顶点图都有一个具有$O(n^{3/2})$体积的三维网格图。这是最广泛的一类图表承认这样的图纸。有向图的三维网格图是\emph{upward}，如果每个弧都指向z方向。证明了每个有向无圈图都有一个体积为$(n^3)$的向上三维网格图，且该图对完全图是紧的。之前的最佳上限是$O(n^4)$。我们的主要结果是：每个$c$ -列有向无圈图（$c$常数）都有一个向上的三维网格图，其体积为$O(n^2)$。这一结果匹配的无向情况下的界限，并提高了最好的已知的界限从$O(n^3)$为许多类的有向无环图，包括平面，串联平行，和outerplanar。
英文摘要：
A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar.
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