Non-separating subgraphs after deleting many disjoint paths

Motivated by the well-known conjecture by Lovász [6] on the connectivity after the path removal, we study the following problem: There exists a function f = f(k, l) such that the following holds. For every f(k, l)connected graph G and two distinct vertices s and t in G, there are k internally disjoint paths P1, . . . , Pk with endpoints s and t such that G− ⋃k i=1 V (Pi) is l-connected. When k = 1, this problem corresponds to Lovász conjecture, and it is open for all the cases l ≥ 3. We show that f(k, 1) = 2k+1 and f(k, 2) ≤ 3k+2. The connectivity “2k+1” for f(k, 1) is best possible. Thus our result generalizes the result by Tutte [8] for the case k = 1 and l = 1 (the first settled case of Lovász conjecture), and the result by Chen, Gould and Yu [1], Kriesell [4], Kawarabayashi, Lee, and Yu [2], independently, for the case k = 1 and l = 2 (the second settled case of Lovász conjecture). When l = 1, our result also improves the connectivity bound “22k + 2” given by Chen, Gould and Yu [1]. June 2009, revised October 11, 2010.