2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76331We 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.10 pages, 2 figuresCombinatorics05C15 (Primary) 05C75, 05C25 (Secondary)text