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

Gyárfás et¿al. and Zaker have proven that the Grundy number of a graph G satisfies ¿ ( G ) ¿ 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 ¿ ( G ) ¿ t and ¿ ¿ ( G ) ¿ 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.