# Parameterized Pre-coloring Extension and List Coloring Problems

## TL;DR

This paper investigates the parameterized complexity of list coloring problems with clique modulator parameters, providing new algorithms and kernelization results that improve understanding and efficiency in solving these graph coloring problems.

## Contribution

It introduces an $O^*(2^k)$ randomized algorithm for list coloring with clique modulators and develops polynomial kernels for related problems, advancing parameterized complexity analysis.

## Key findings

- An $O^*(2^k)$ randomized algorithm for list coloring with clique modulators.
- A kernel with at most $3k$ vertices for list coloring extension.
- A polynomial kernel with $O(k^2)$ vertices and colors for a related list coloring problem.

## Abstract

Golovach, Paulusma and Song (Inf. Comput. 2014) 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 $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the 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 $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.