A note on Lovasz removable path conjecture

Lovasz [8] conjectured that for any natural number k, there exists a smallest natural number f(k) such that, for any two vertices s and t in any f(k)-connected graph G, there exists an s-t path P such that G - V(P) is k-connected. This conjecture is proved only for k <= 2. Here, we strengthen the result for k = 2 as follows: for any integers l > 0 and m >= 0, there exists a function f(l, m) such that, for any distinct vertices s, t, v(1),..., v(m) in any f(l, m)-connected graph G, there exist l internally vertex disjoint s-t paths P-1,..., P-l such that for any subset I subset of{1,..., l}, G - boolean OR V-i is an element of I(P-i) is 2-connected and {v(1), v(2),..., v(m)} subset of V (G) - boolean OR V-1 <= i <= l(P-i).