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Parameterized Pre-Coloring Extension and List Coloring Problems
SIAM Journal on Discrete Mathematics ( IF 0.9 ) Pub Date : 2021-03-29 , DOI: 10.1137/20m1323369
Gregory Gutin , Diptapriyo Majumdar , Sebastian Ordyniak , Magnus Wahlström

SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 575-596, January 2021.
Golovach, Paulusma, and Song [Inform. and Comput., 237 (2014), pp. 204--214] asked to determine the parameterized complexity of the following problems parameterized by $k$: 1. Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whether $G$ has a proper list coloring. 2. Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $\lambda_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $\lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q$. For problem 1 we design an ${\mathcal O}^*(2^k)$-time randomized algorithm and for problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. [in Proceedings of IWOCA 2019, Springer, Berlin, 2019, pp. 61--69] proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v \in V(G),$ decide whether there is a proper list coloring for $G$. We obtain a kernel with ${\mathcal O}(k^2)$ vertices and colors and a compression to a variation of the problem with ${\mathcal O}(k)$ vertices and ${\mathcal O}(k^2)$ colors.


中文翻译:

参数化预着色扩展和列表着色问题

SIAM 离散数学杂志,第 35 卷,第 1 期,第 575-596 页,2021 年 1 月。
Golovach、Paulusma 和 Song [通知。and Comput., 237 (2014), pp. 204--214] 要求确定由 $k$ 参数化的以下问题的参数化复杂度: 1. 给定图 $G$,一个团调制器 $D$(一个团modulator 是一组顶点,其删除导致 $G$ 的大小为 $k$ 的团),以及每个 $v\in V(G)$ 的颜色列表 $L(v)$,决定 $ G$ 有一个适当的列表着色。2. 给定一个图 $G$,一个大小为 $k$ 的集团调制器 $D$ 用于 $G$,以及一个预着色 $\lambda_P: X \rightarrow Q$ for $X \subseteq V(G),$决定是否可以仅使用 $Q$ 的颜色将 $\lambda_P$ 扩展为 $G$ 的正确着色。对于问题 1,我们设计了一个 ${\mathcal O}^*(2^k)$-time 随机算法,对于问题 2,我们获得了一个最多具有 $3k$ 个顶点的核。巴尼克等人。[在 IWOCA 2019 年会刊,斯普林格,柏林,2019 年,pp. 61--69] 证明了以下问题是固定参数可处理的,并询问它是否承认多项式核:给定一个图 $G$、一个整数 $k$ 和一个正好为 $ 的列表 $L(v)$每个 $v \in V(G),$ 的 nk$ 颜色决定是否有适当的 $G$ 列表着色。我们获得了一个具有 ${\mathcal O}(k^2)$ 顶点和颜色的内核,以及对 ${\mathcal O}(k)$ 顶点和 ${\mathcal O}(k ^2)$ 颜色。
更新日期:2021-03-29
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