Parameterized Pre-Coloring Extension and List Coloring Problems

Abstract

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the param5 eterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique 6 modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, 7 and a list L(v) of colors for every v ∈ V (G), decide whether G has a proper list coloring; (2) Given a 8 graph G, a clique modulator D of size k for G, and a pre-coloring λP : X → Q for X ⊆ V (G), decide 9 whether λP can be extended to a proper coloring of G using only colors from Q. For Problem 1 we 10 design an O∗(2k)-time randomized algorithm and for Problem 2 we obtain a kernel with at most 11 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable 12 and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) 13 of exactly n− k colors for every v ∈ V (G), decide whether there is a proper list coloring for G. We 14 obtain a kernel with O(k2) vertices and colors and a compression to a variation of the problem with 15 O(k) vertices and O(k2) colors. 16