The recognition of geodetically connected graphs

✍ Scribed by Jou-Ming Chang; Chin-Wen Ho

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 849 KB
• Volume: 65
• Category: Article
• ISSN: 0020-0190
• DOI: 10.1016/s0020-0190(97)00201-9

✦ Synopsis

Let G = ( y E) be a graph with vertex set V of size n and edge set E of size m. A vertex LJ E V is called a hinge vertex if the distance of any two vertices becomes longer after u is removed. A graph without hinge vertex is called a hinge-free graph. In general, a graph G is k-geodetically connected or k-GC for short if G can tolerate any k -1 vertices failures without increasing the distance among all the remaining vertices. In this paper, we show that recognizing a graph G to be k-GC for the largest value of k can be solved in O(nm) time. In addition, more efficient algorithms for recognizing the k-GC 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 0( 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 hinge-free then finding all hinge vertices of G can be solved in the same time complexity on the above classes of graphs. @