, Volume 23, Issue 2, pp 185203
First online:
Graph Connectivity After Path Removal
 Guantao Chen*Affiliated withDept. of Math/Stat., Georgia State University Email author
 , Ronald J. Gould†Affiliated withDepartment of Math/CS, Emory University
 , Xingxing Yu‡Affiliated withSchool of Mathematics, Georgia Tech
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Get AccessLet G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer α(k) so that if G is an α(k)connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P _{1}[u,v], P _{2}[u,v], . . ., P _{ k }[u,v], such that G—V (P _{ i }[u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that α(k) ≤ 22k+2. It is known that α(1) = 3.. A result of Tutte showed that α(2) = 3. We show that α(3) = 6. In addition, we prove that if G is a 5connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2connected.
AMS Subject Classification (2000):
05C40 05C38 Title
 Graph Connectivity After Path Removal
 Journal

Combinatorica
Volume 23, Issue 2 , pp 185203
 Cover Date
 200304
 DOI
 10.1007/s0030018z
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Keywords

 05C40
 05C38
 Industry Sectors
 Authors

 Guantao Chen* ^{(1)}
 Ronald J. Gould† ^{(2)}
 Xingxing Yu‡ ^{(3)}
 Author Affiliations

 1. Dept. of Math/Stat., Georgia State University, Atlanta, GA, 30303, USA
 2. Department of Math/CS, Emory University, Atlanta, GA, 30322, USA
 3. School of Mathematics, Georgia Tech, Atlanta, GA, 30332, USA