Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4646999 | Discrete Mathematics | 2016 | 11 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Brice Effantin, Nicolas Gastineau, Olivier Togni,