  We show that for all simple graphs G other than the cycles C_3,C_4,C_5, and
the claw K_1,3 there exists a K > 0 such that whenever k > K the k-th iterate
of the line graph can be distinguished by at most two colors. Additionally we
determine, for trees, when the distinguishing number of the line graph of T is
greater than the distinguishing number of T.
