We study the computational complexity of $c$-Colored $P_\ell$ Deletion and $C_\ell$ Deletion. In these problems, one is given a $c$-edge-colored graph wants to destroy all induced $c$-colored paths or cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein, path cycle if it contains edges $c$ distinct colors. show that are NP-hard for each non-trivial combination $\ell$. ...