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Algorithm for Identifying the Maximum Detour Hinge Vertices of a Permutation Graph
Hirotoshi HONMA Yoko NAKAJIMA Yuta IGARASHI Shigeru MASUYAMA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E98A
No.6
pp.11611167 Publication Date: 2015/06/01
Online ISSN: 17451337
DOI: 10.1587/transfun.E98.A.1161
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications) Category: Keyword: design and analysis of algorithms, intersection graphs, maximum detour hinge vertex problem, permutation graph,
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Summary:
A hinge vertex is a vertex in an undirected graph such that there exist two vertices whose removal makes the distance between them longer than before. Identifying hinge vertices in a graph can help detect critical nodes in communication network systems, which is useful for making them more stable. For finding them, an O(n^{3}) time algorithm was developed for a simple graph, and, linear time algorithms were developed for interval and permutation graphs, respectively. Recently, the maximum detour hinge vertex problem is defined by Honma et al. For a hinge vertex u in a graph, the detour degree of u is the largest value of distance between any pair of x and y (x and y are adjacent to u) by removing u. A hinge vertex with the largest detour degree in G is defined as the maximum detour hinge vertex of G. This problem is motivated by practical applications, such as network stabilization with a limited cost, i.e., by enhancing the reliability of the maximum detour hinge vertex, the stability of the network is much improved. We previously developed an O(n^{2}) time algorithm for solving this problem on an interval graph. In this study, we propose an algorithm that identifies the maximum detour hinge vertex on a permutation graph in O(n^{2}) time, where n is the number of vertices in the graph.

