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.