Non-separating subgraphs after deleting many disjoint paths

Motivated by the well-known conjecture by Lovász (1975) [6] on the connectivity after the path removal, we study the following problem: 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 P 1 , … , P k with endpoints s and t such that G − ⋃ i = 1 k V ( P i ) is l-connected. = 1 , this problem corresponds to Lovász conjecture, and it is open for all the cases l ⩾ 3 . w that f ( k , 1 ) = 2 k + 1 and f ( k , 2 ) ⩽ 3 k + 2 . The connectivity “ 2 k + 1 ” for f ( k , 1 ) is best possible. Thus our result generalizes the result by Tutte (1963) [8] for the case k = 1 and l = 1 (the first settled case of Lovász conjecture), and the result by Chen, Gould and Yu (2003) [1], Kriesell (2001) [4], Kawarabayashi, Lee, and Yu (2005) [2], independently, for the case k = 1 and l = 2 (the second settled case of Lovász conjecture). = 1 , our result also improves the connectivity bound “ 22 k + 2 ” given by Chen, Gould and Yu (2003) [1].