{"paper":{"title":"A characterization of b-chromatic and partial Grundy numbers by induced subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Brice Effantin (GOAL), Nicolas Gastineau (Le2i), Olivier Togni (Le2i)","submitted_at":"2015-05-28T18:05:28Z","abstract_excerpt":"Gy{\\'a}rf{\\'a}s et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\\Gamma(G)\\ge t$ 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 $\\varphi(G)\\ge t$ and $\\partial\\Gamma(G)\\ge t$ (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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07780","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}