Fast FPT Algorithms for Grundy Number on Dense Graphs

In this paper, we investigate the Grundy Coloring problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the first-fit coloring of a graph in the worstcase vertex ordering, is known to be W[1]-hard when parameterized by the number of colors required by the most adversarial ordering. We focus on fixed-parameter tractable (FPT) algorithms for solving this problem on graph classes characterized by dense substructures, specifically those with a cluster modulator. A cluster modulator is a vertex subset whose removal results in a cluster graph (a disjoint union of cliques). We present FPT algorithms for graphs where the cluster graph consists of one, two, or k cliques, leveraging the cluster modulator’s properties to achieve the best-known FPT runtimes, parameterized by both the modulator’s size and the number of cliques.