Destroying Multicolored Paths and Cycles in Edge-Colored Graphs

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 $\ell$. then analyze parameterized problems. extend notion neighborhood diversity edge-colored graphs both problems fixed-parameter tractable with respect colored input graph. also provide hardness results outline limits parameterization standard parameter solution size $k$. Finally, we consider bicolored special case $2$-Colored $P_4$ can be solved in polynomial time.