  Let F_2 denote the free group of rank 2. Our main technical result of
independent interest is: for any element u of F_2, there is g in F_2 such that
no cyclically reduced image of u under an automorphism of F_2 contains g as a
subword. We then address computational complexity of the following version of
the Whitehead automorphism problem: given a fixed u in F_2, decide, on an input
v in F_2 of length n, whether or not v is an automorphic image of u. We show
that there is an algorithm that solves this problem and has constant (i.e.,
independent of n) average-case complexity.
