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.

中文翻译：

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销毁边缘彩色图中的五彩路径和循环

我们研究了$ c $彩色$ P_ \ ell $删除和$ c $彩色$ C_ \ ell $删除的计算复杂度。在这些问题中，给了一个$ c $边彩色的图，并希望通过删除最多$ k $边来破坏$ \ ell $顶点上所有诱导的$ c $彩色路径或循环。在此，如果路径或循环包含$ c $不同颜色的边缘，则该路径或循环为$ c $颜色。我们显示，对于$ c $和$ \ ell $的每个非平凡组合，$ c $彩色$ P_ \ ell $删除和$ c $彩色$ C_ \ ell $删除都是NP难的。然后，我们分析这些问题的参数化复杂性。我们将邻域多样性的概念扩展到边缘彩色图，并表明相对于输入图的彩色邻域多样性，这两个问题都是固定参数可处理的。我们还提供硬度结果，以概述标准参数解决方案大小$ k $的参数化限制。最后，我们考虑双色输入图，并显示了可以在多项式时间内求解的$ 2 $ -Colored $ P_4 $ Deletion的特殊情况。