A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

TL;DR

This paper introduces new characterizations of b-chromatic and partial Grundy numbers using induced subgraphs called t-atoms, providing complexity results and applications in graph theory.

Contribution

It extends the concept of t-atoms to b-coloring and partial Grundy coloring, offering new tools for analyzing these parameters.

Findings

Determining if (G) er t is in XP with parameter t.

Results on b-critical vertices and edges.

Insights into b-perfect graphs and graphs with girth at least 7.

Abstract

Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\Gamma(G)\ge t$ if and only if $G$ contains an induced subgraph called a $t$-atom.The family of $t$-atoms has bounded order and contains a finite number of graphs.In this article, we introduce equivalents of $t$-atoms for b-coloring and partial Grundy coloring.This concept is used to prove that determining if $\varphi(G)\ge t$ and $\partial\Gamma(G)\ge t$ (under conditions for the b-coloring), for a graph $G$, is in XP with parameter $t$.We illustrate the utility of the concept of $t$-atoms by giving results on b-critical vertices and edges, on b-perfect graphs and on graphs of girth at least $7$.