Gutin, G, Majumdar, D, Ordyniak, S orcid.org/0000-0003-1935-651X et al. (1 more author) (2021) Parameterized Pre-Coloring Extension and List Coloring Problems. SIAM Journal on Discrete Mathematics, 35 (1). pp. 575-596. ISSN 0895-4801
Abstract
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021, Society for Industrial and Applied Mathematics. This is an author produced version of an article published in SIAM Journal on Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | parameterized complexity; kernelization; graph coloring; list coloring |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 18 Jun 2021 11:57 |
Last Modified: | 18 Jun 2021 11:57 |
Status: | Published |
Publisher: | Society for Industrial & Applied Mathematics (SIAM) |
Identification Number: | 10.1137/20m1323369 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:175347 |