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 Γ(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 tatoms for b-coloring and partial Grundy coloring. This concept is used to prove that determining if φ(G) ≥ t and ∂Γ(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.