# Graph Connectivity After Path Removal

- Received:

DOI: 10.1007/s003-0018-z

- Cite this article as:
- Chen*, G., Gould†, R.J. & Yu‡, X. Combinatorica (2003) 23: 185. doi:10.1007/s003-0018-z

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*) ≤ 22*k*+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.