平面三角剖分的有向直径
Oriented Diameter of Planar Triangulations
摘要：
无向图或有向图的直径被定义为图中所有顶点对上的最大最短路径距离。给定一个无向图$G$，我们研究的问题，分配方向的$G$的每一个边缘，使所得到的有向图的直径最小化。所有强连通方向上的最小直径称为$G$的定向直径。确定图的定向直径的问题是NP-难的，但时间复杂性问题对于平面图是开放的。本文给出了三角网格图定向直径的精确计算方法。然后，我们证明了一个$n/3$下界和一个$n/2+O(\sqrt{n})$上界的平面三角剖分的有向直径。已知给定一个树宽有界的平面图$G$和一个固定的正整数$k$，可以在线性时间内判定$G$的定向直径是否至多为$k$ .相反，我们考虑了一个加权版本的定向直径问题，并证明它是弱NP-完全的平面图有界路宽。
英文摘要：
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$ such that the diameter of the resulting oriented graph is minimized. The minimum diameter over all strongly connected orientations is called the oriented diameter of $G$. The problem of determining the oriented diameter of a graph is known to be NP-hard, but the time-complexity question is open for planar graphs. In this paper we compute the exact value of the oriented diameter for triangular grid graphs. We then prove an $n/3$ lower bound and an $n/2+O(\sqrt{n})$ upper bound on the oriented diameter of planar triangulations. It is known that given a planar graph $G$ with bounded treewidth and a fixed positive integer $k$, one can determine in linear time whether the oriented diameter of $G$ is at most $k$. In contrast, we consider a weighted version of the oriented diameter problem and show it to be is weakly NP-complete for planar graphs with bounded pathwidth.
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