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，目标是通过删除至多$k$条边，破坏所有诱导的$c$色路径或$c$色圈（每个含$\ell$个顶点）。其中，路径或圈若包含$c$种不同颜色的边，则称为$c$色。我们证明，对于$c$和$\ell$的每个非平凡组合，$c$-Colored $P_\ell$ Deletion和$c$-Colored $C_\ell$ Deletion均为NP难问题。随后，我们分析这些问题的参数化复杂性。我们将邻域多样性的概念扩展到边着色图，并证明这两个问题相对于输入图的有色邻域多样性具有固定参数可解性。我们还提供困难性结果，以说明通过标准参数解大小$k$进行参数化的局限性。最后，我们考虑双色输入图，并证明$2$-Colored $P_4$ Deletion的一个特殊情形可在多项式时间内求解。