Fast FPT Algorithms for Grundy Number on Dense Graphs

TL;DR

This paper develops fixed-parameter tractable algorithms for the Grundy coloring problem on dense graphs with a cluster modulator, improving computational efficiency for certain dense graph classes.

Contribution

It introduces new FPT algorithms for Grundy coloring on graphs with a cluster modulator, especially those with a small number of cliques, advancing the understanding of coloring in dense graphs.

Findings

FPT algorithms for graphs with one, two, or k cliques in the cluster graph

Achieved the best-known FPT runtimes parameterized by modulator size and number of cliques

Extended the applicability of Grundy coloring algorithms to dense graph classes

Abstract

In this paper, we investigate the \textsc{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 worst-case 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.