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:在这些问题中,我们给出了一个以美元为顶尖的彩色图表, 并想要通过删除大部分以美元为底端的边緣来摧毁所有由美元引发的彩色路径或周期。 这里, 路径或周期是以美元为底色的, 如果含有以美元为底色的边缘, 则以美元为底色的。 我们显示, 美元为美元为底色的彩色和 美元为美元为底色的彩色。 在这些问题中, 给一个人一个以美元为底色的彩色图表, 并想要用美元和美元为美元的非三维组合来摧毁所有由美元引发的彩色路径或周期。 我们然后分析这些问题的参数复杂性。 我们把社区多样性的概念扩展到以色色的图形, 并表明, 两种问题都是固定的参数, 与彩色的社区多样性有关。 我们还提供了硬度结果, 以美元为底色的美元为底色的基数, 能够用标准基数的硬度标度来勾划出标准基数的基数 。