{"paper":{"title":"Fire Containment in Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"F\\'elix Sipma, Fr\\'ed\\'eric Maffray, Jan van den Heuvel, Louis Esperet","submitted_at":"2011-02-15T09:55:38Z","abstract_excerpt":"In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $\\rho_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs $\\cG$ we are interested in the minimum value $k$ such that $\\rho_k(G)\\ge\\epsilon$ for some constant $\\epsilon>0$ and all $G\\in\\cG$ i.e., such that linearly many vertices are expected to be saved in every graph from"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3016","kind":"arxiv","version":4},"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"}