Abstract Lovasz 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 ) ≤ 2 k + 1 .