Path extendability of claw-free graphs

Let GG be a connected, locally connected, claw-free graph of order nn and x,yx,y be two vertices of GG. In this paper, we prove that if for any 2-cut SS of GG, S∩{x,y}=∅S∩{x,y}=∅, then each (x,y)(x,y)-path of length less than n-1n-1 in GG is extendable, that is, for any path PP joining xx and yy of length h(