{"paper":{"title":"Perfect domination in regular grid graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Italo J. Dejter","submitted_at":"2007-11-27T20:40:28Z","abstract_excerpt":"We show there is an uncountable number of parallel total perfect codes in the integer lattice graph ${\\Lambda}$ of $\\R^2$. In contrast, there is just one 1-perfect code in ${\\Lambda}$ and one total perfect code in ${\\Lambda}$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of ${\\Lambda}$ and $C_m\\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\\Z_{2d^2+2d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4343","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"}