A Study on Metric Properties of Graphs: Hinge Vertices and Graph Powers

• Main Authors: Ting-Yem Ho, 何定燕
• Other Authors: Yue-Li Wang
• Format: Others
• Language: en_US
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/07312724064602441825

博士 === 國立臺灣科技大學 === 資訊管理系 === 88 === Abstract In this dissertation, two topics related to the metric properties of graphs are considered. One is the hinge vertex and the other is the graph power. The first part of this dissertation looks at the algorithmic complexities of the problem for finding all hinge vertices of a graph. A vertex v in a graph G is called a hinge vertex if there exist two other vertices with distance becoming longer after v is removed. This means that in the corresponding network of G, the fault of a hinge vertex will increase the communication cost to the remaining network. Thus, finding all hinge vertices is an important problem for network analysis, which can be used to identify critical nodes in a real network. We then develop two algorithms for solving this problem on the special graphs, called permutation graphs and strongly chordal graphs. For permutation graphs, we propose an O(n) time algorithm if a permutation diagram is provided. For strongly chordal graphs, we propose an O(n+m) time algorithm if a strong elimination ordering is given. The second part of this dissertation deals with the topic on graph power, specifically, the closure property of graph classes under power operation. Many results obtained from our study are related to the graphs with bounded asteroidal number. The asteroidal number of a graph is the maximum cardinality of a vertex set A such that for every vertex aÎA, if the removal of the closed neighborhood of a in the graph, the set A\{a} is contained in one connected component of the remaining graph. We prove that every class of graphs with bounded asterodial number is closed under power. This generalizes a previous result for the family of AT-free graphs. Moreover, we consider the forbidden configurations for the powers of graphs with bounded asteroidal number. Based on these forbidden configurations, we show that every partition problem, called k-clustering with k³2, can be solved in polynomial time on AT-free graphs.