TY - JOUR

T1 - The recognition of geodetically connected graphs

AU - Changa, Jou Ming

AU - Ho, Chin Wen

N1 - Funding Information:
* Corresponding author. Email: hocw@csie.ncu.edu.hv. 1T his work was supported by the National Science Council under the Contract NSC87-2213-E-008-017.

PY - 1998

Y1 - 1998

N2 - Let G = (V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v ∈ 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 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 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 hinge-free then finding all hinge vertices of G can be solved in the same time complexity on the above classes of graphs.

AB - Let G = (V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v ∈ 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 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 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 hinge-free then finding all hinge vertices of G can be solved in the same time complexity on the above classes of graphs.

KW - Geodetically connected

KW - Hinge vertices

KW - Recognition algorithm

UR - http://www.scopus.com/inward/record.url?scp=0038246870&partnerID=8YFLogxK

U2 - 10.1016/s0020-0190(97)00201-9

DO - 10.1016/s0020-0190(97)00201-9

M3 - 期刊論文

AN - SCOPUS:0038246870

VL - 65

SP - 81

EP - 88

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 2

ER -