Non-separating subgraphs

Lovász conjectured that there is a smallest integer f (l) such that for every f (l)-connected graph G and every two vertices s, t of G there is a path P connecting s and t such that G − V (P) is l-connected. This conjecture is still open for l ≥ 3. In this paper, we generalize this conjecture to a k-vertex version: is there a smallest integer f (k, l) such that for every f (k, l)-connected graph and every subset X with k vertices, there is a tree T connecting X such that G − V (T ) is l-connected? We prove that f (k, 1) = k + 1 and f (k, 2) ≤ 2k + 1. © 2012 Elsevier B.V. All rights reserved.