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.

翻译：我们研究 $c$-Colored $P_\ell$ Deletion（$c$色$P_\ell$删除）和 $c$-Colored $C_\ell$ Deletion（$c$色$C_\ell$删除）的计算复杂性。在这些问题中，给定一个 $c$边着色图，要求通过删除至多 $k$ 条边，摧毁所有在 $\ell$ 个顶点上诱导的 $c$色路径或环。其中，若一条路径或环包含 $c$ 种不同颜色的边，则称其为 $c$色的。我们证明，对于 $c$ 和 $\ell$ 的每种非平凡组合，$c$-Colored $P_\ell$ Deletion 和 $c$-Colored $C_\ell$ Deletion 均为 NP-难问题。随后分析这些问题的参数化复杂性。我们将邻域多样性概念推广至边着色图，并证明这两个问题关于输入图的着色邻域多样性具有固定参数可解性。我们还提供难解性结果，以说明标准参数解规模 $k$ 的参数化局限性。最后，考虑双色输入图，并证明 $2$-Colored $P_4$ Deletion 的一个特例可在多项式时间内求解。