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引用次数: 0
摘要
无向图或有向图的直径定义为图中所有顶点对的最大最短路径距离。给定一个无向图 G,我们要研究的问题是为 G 的每条边分配方向,从而使生成的有向图的直径最小。在所有强连接方向上的最小直径称为 G 的有向直径。确定图的有向直径问题是已知的 NP 难问题,但对于平面图来说,时间复杂性问题尚未解决。在本文中,我们计算了三角形网格图的定向直径的精确值。然后,我们证明了平面三角形的定向直径的 n/3 下限和 \(n/2+O\left( \sqrt{n}\right) \) 上限,其中 n 是 G 中的顶点数。众所周知,给定一个有界树宽、固定正整数 k 的平面图 G,我们可以在线性时间内确定 G 的定向直径是否最多为 k。我们考虑了定向直径问题的加权版本,并证明它对于有界路径宽的平面图来说是弱 NP-完全的。
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 \(n/2+O\left( \sqrt{n}\right) \) upper bound on the oriented diameter of planar triangulations, where n is the number of vertices in G. 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. We consider a weighted version of the oriented diameter problem and show it to be weakly NP-complete for planar graphs with bounded pathwidth.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.