三正则平面图的有效重组
Efficient Reassembling of Three-Regular Planar Graphs
摘要：
简单图G =（V，E）的重组是对早期网络分析研究中出现的一个问题的抽象。图重组有几个等效的定义;在本报告中，我们使用了一个最接近图雕刻概念的定义。一个重组是一个根二叉树，它的节点是V的子集，它的叶节点是单例集，每个单例集包含G的一个不同的顶点。重组中两个节点的父节点是两个子节点顶点集的并集。重组的根结点是全集合V。重组中的结点的边边界度是G中连接结点集合中的顶点到不在结点集合中的顶点的边的数量。一个重组的alpha测度是重组中任何节点的最大边边界度。G的重组是α-最优的，如果它的α-测度是G的重组的所有α-测度中的最小值。找到一个简单图的阿尔法最优重组的问题一般已经被证明是NP困难的。本文提出了一个算法，给定一个3-正则平面图G =（V，E）作为输入，该算法返回一个α测度与n（G的顶点数）无关且上界为2k的G的重组图，其中k是G的边外平面度。（边外平面性是不同的，但与通常的外平面性概念密切相关;与外平面性一样，对于固定的边外平面性k，顶点数n可以是任意大的。我们的算法运行在时间线性n。此外，我们构造了一类$3$ -正则平面图，这种α-措施是最佳的，通过证明，2k是下界的α-措施的任何重组图在该类。
英文摘要：
A reassembling of a simple graph G = (V,E) is an abstraction of a problem arising in earlier studies of network analysis. There are several equivalent definitions of graph reassembling; in this report we use a definition which makes it closest to the notion of graph carving. A reassembling is a rooted binary tree whose nodes are subsets of V and whose leaf nodes are singleton sets, with each of the latter containing a distinct vertex of G. The parent of two nodes in the reassembling is the union of the two children's vertex sets. The root node of the reassembling is the full set V. The edge-boundary degree of a node in the reassembling is the number of edges in G that connect vertices in the node's set to vertices not in the node's set. A reassembling's alpha-measure is the largest edge-boundary degree of any node in the reassembling. A reassembling of G is alpha-optimal if its alpha-measure is the minimum among all alpha-measures of G's reassemblings. The problem of finding an alpha-optimal reassembling of a simple graph in general was already shown to be NP-hard. In this report we present an algorithm which, given a 3-regular plane graph G = (V,E) as input, returns a reassembling of G with an alpha-measure independent of n (number of vertices in G) and upper-bounded by 2k, where k is the edge-outerplanarity of G. (Edge-outerplanarity is distinct but closely related to the usual notion of outerplanarity; as with outerplanarity, for a fixed edge-outerplanarity k, the number n of vertices can be arbitrarily large.) Our algorithm runs in time linear in n. Moreover, we construct a class of $3$-regular plane graphs for which this alpha-measure is optimal, by proving that 2k is the lower bound on the alpha-measure of any reassembling of a graph in that class.
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