{"paper":{"title":"Graph functionality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aistis Atminas, Bogdan Alecu, Vadim Lozin","submitted_at":"2018-07-04T19:17:28Z","abstract_excerpt":"Let $G=(V,E)$ be a graph and $A$ its adjacency matrix. We say that a vertex $y \\in V$ is a function of vertices $x_1, \\ldots, x_k \\in V$ if there exists a Boolean function $f$ of $k$ variables such that for any vertex $z \\in V - \\{y, x_1, \\ldots, x_k\\}$, $A(y,z)=f(A(x_1,z),\\ldots,A(x_k,z))$. The functionality $fun(y)$ of vertex $y$ is the minimum $k$ such that $y$ is a function of $k$ vertices. The functionality $fun(G)$ of the graph $G$ is $\\max\\limits_H\\min\\limits_{y\\in V(H)}fun(y)$, where the maximum is taken over all induced subgraphs $H$ of $G$. In the present paper, we show that function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.01749","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"}