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

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• 作者：Brice Effantin^b ; ^{brice.effantin-dit-toussaint@univ-lyon1.fr" class="auth_mail" title="E-mail the corresponding author} ; Nicolas Gastineau^a ; ^{Nicolas.Gastineau@u-bourgogne.fr" class="auth_mail" title="E-mail the corresponding author} ; Olivier Togni^a ; ^{Olivier.Togni@u-bourgogne.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Parameterized complexity ; Grundy number ; b-coloring ; Partial Grundy number
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 August 2016
• 年：2016
• 卷：339
• 期：8
• 页码：2157-2167
• 全文大小：460 K

文摘

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