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 P1[u,v], P2[u,v], . . ., Pk[u,v], such that G—V (Pi[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.