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 GG satisfies Γ(G)≥tΓ(G)≥t if and only if GG contains an induced subgraph called a tt-atom. The family of tt-atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of tt-atoms for b-coloring and partial Grundy coloring. This concept is used to prove that determining if φ(G)≥tφ(G)≥t and ∂Γ(G)≥t∂Γ(G)≥t (under conditions for the b-coloring), for a graph GG, is in XP with parameter tt. We illustrate the utility of the concept of tt-atoms by giving results on b-critical vertices and edges, on b-perfect graphs and on graphs of girth at least 7.