找到少量有颜色的成分
Finding a Small Number of Colourful Components
摘要：
如果图$G$的顶点集的任何一个$V_i$中没有两个顶点具有相同的颜色，并且每个集合$V_i$都导出一个连通图，则图$G$的顶点集的一个带有（不一定是适当的）着色$c$的划分$(V_1,\ldots,V_k)$是有色的.彩色划分问题是判定一个彩色图$(G,c)$是否有一个大小至多为$k$的彩色划分。这个问题与色分量问题密切相关，色分量问题是指通过删除最多$p$边来决定一个图是否可以被修改成其连通分量形成一个色分区的图。然而，我们表明，色划分和色成分可能有不同的复杂性，为有限的情况。我们收紧已知的NP-硬度的结果，这两个问题，此外，我们证明了新的硬度和tractability结果的彩色分割。利用这些结果，我们完成了我们的论文，对彩色分割进行了彻底的参数化研究。
英文摘要：
A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most $p$ edges. Nevertheless we show that COLOURFUL PARTITION and COLOURFUL COMPONENTS may have different complexities for restricted instances. We tighten known NP-hardness results for both problems and in addition we prove new hardness and tractability results for COLOURFUL PARTITION. Using these results we complete our paper with a thorough parameterized study of COLOURFUL PARTITION.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
