{"paper":{"title":"The Domination Equivalence Classes of Paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Iain Beaton, Jason I. Brown","submitted_at":"2017-10-11T01:08:28Z","abstract_excerpt":"A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $\\gamma (G)$, is the cardinality of the smallest dominating set of $G$. The domination polynomial is defined by $D(G,x) = \\sum d(G,i)x^i$ where $d(G,i)$ is the number of dominating sets in $G$ with cardinality $i$. Two graphs $G$ and $H$ are considered $\\mathcal{D}$-equivalent if $D(G,x)=D(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\\mathcal{D}$-equivalent to $G$. Extendi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.03871","kind":"arxiv","version":1},"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"}