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The Recognition of Geodetically Connected Graphs
, 1998
"... Let G = (V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v 2 V is called a hinge vertex if the distance of any two vertices becomes longer after v is removed. A graph without hinge vertex is called a hingefree graph. In general, a graph G is kgeodetically connected ..."
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Let G = (V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v 2 V is called a hinge vertex if the distance of any two vertices becomes longer after v is removed. A graph without hinge vertex is called a hingefree graph. In general, a graph G is kgeodetically connected or kGC for short if G can tolerate any k,1vertices failures without increasing the distance among all the remaining vertices. In this paper, we show that recognizing a graph G to be kGC for the largest value of k can be solved in O(nm) time. In addition, more efficient algorithms for recognizing the kGC property on some special graphs are presented. These include the O(n+m) time algorithms on strongly chordal graphs (if a strong elimination ordering is given), ptolemaic graphs, and interval graphs, and an O(n²) time algorithm on undirected path graphs (if a characteristic tree model is given). Moreover, we show that if the input graph G is not hingefree then finding all hinge vertices of ...
RECOGNIZING HINGEFREE LINE GRAPHS AND TOTAL GRAPHS
"... Abstract. In this paper, we characterize line graphs and total graphs that are hingefree, i.e., there is no triple of verticesx;y;z such that the distance betweeny andz increases afterx is removed. Based on our characterizations, we show that given a graph G with n vertices andm edges, determining ..."
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Abstract. In this paper, we characterize line graphs and total graphs that are hingefree, i.e., there is no triple of verticesx;y;z such that the distance betweeny andz increases afterx is removed. Based on our characterizations, we show that given a graph G with n vertices andm edges, determining its line graph and total graph to be hingefree can be solved in O(n +m) time. Moreover, characterizations of hingefree iterated line graphs and total graphs are also discussed. 1.
Finding the Set of All Hingevertices for Strongly Chordal Graphs in Linear Time
"... Let V (G) be the vertex set of a graph G = (V; E) and G \Gamma u be the subgraph induced by the vertex set V (G) \Gamma fug. A vertex u 2 V (G) is said to be a hingevertex of G if and only if there exist two vertices x; y 2 V (G \Gamma u) such that dG\Gammau (x; y) ? dG (x; y), where dG (x; y) is t ..."
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Let V (G) be the vertex set of a graph G = (V; E) and G \Gamma u be the subgraph induced by the vertex set V (G) \Gamma fug. A vertex u 2 V (G) is said to be a hingevertex of G if and only if there exist two vertices x; y 2 V (G \Gamma u) such that dG\Gammau (x; y) ? dG (x; y), where dG (x; y) is the distance (the length of a shortest path) between x and y in G. In this paper, based on the breadth first search technique, an O(n + e) time algorithm is proposed for finding the set of all hingevertices of a strongly chordal graph when a strong elimination ordering is given. 1. INTRODUCTION Most of network designs and analysis usually model their topologies as graphical representations in a natural way because that many relevant problems of networks can be solved by using graph theoretic results. As usual, a communication network is modeled as an undirected graph, in which nodes and edges correspond to the communication sites and links, respectively. For the distributed networks, the c...
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"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: