  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 $7$.
