Parameterized Pre-Coloring Extension and List Coloring Problems

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