Graph Connectivity After Path Removal

Let 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 5-connected 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 2-connected.

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Guantao Chen*.

Additional information

* Supported by NSF grant No. DMS-0070059

† Supported by ONR grant N00014-97-1-0499

‡ Supported by NSF grant No. 9531824

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen*, G., Gould†, R.J. & Yu‡, X. Graph Connectivity After Path Removal. Combinatorica 23, 185–203 (2003). https://doi.org/10.1007/s003-0018-z

Download citation

AMS Subject Classification (2000):

  • 05C40
  • 05C38