  We study the computational complexity of $c$-Colored $P_\ell$ Deletion and
$c$-Colored $C_\ell$ Deletion. In these problems, one is given a
$c$-edge-colored graph and wants to destroy all induced $c$-colored paths or
cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein,
a path or cycle is $c$-colored if it contains edges of $c$ distinct colors. We
show that $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion are
NP-hard for each non-trivial combination of $c$ and $\ell$. We then analyze the
parameterized complexity of these problems. We extend the notion of
neighborhood diversity to edge-colored graphs and show that both problems are
fixed-parameter tractable with respect to the colored neighborhood diversity of
the input graph. We also provide hardness results to outline the limits of
parameterization by the standard parameter solution size $k$. Finally, we
consider bicolored input graphs and show a special case of $2$-Colored $P_4$
Deletion that can be solved in polynomial time.
