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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.01749","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-04T19:17:28Z","cross_cats_sorted":[],"title_canon_sha256":"3186a59f3bdbc18a03652519b86eb2f204fc1bfb0a7eacceff0461e150eee1cb","abstract_canon_sha256":"89c8619cedf46292cefa7f214c73bb9eda17503c080a3f63110bc83ed6057059"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:28.726282Z","signature_b64":"6okaSw9Ki6iwqz2NMscvep4m85iBPvw3DEPEoVcBiV0ivmAN/LVlenHFHNNGMOGb6i9zuDX/cRFKAiT+0fusBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54d44534e4da6289dcd46504cda334b5ad8c373dbc80c7c7afd3f0a21947407a","last_reissued_at":"2026-05-18T00:11:28.725869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:28.725869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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