{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:DZZAWW3JXOPSXALWS35DOHNGSJ","short_pith_number":"pith:DZZAWW3J","schema_version":"1.0","canonical_sha256":"1e720b5b69bb9f2b817696fa371da6926c665f763d7cf51c713709fbf769275d","source":{"kind":"arxiv","id":"1402.0134","version":2},"attestation_state":"computed","paper":{"title":"On the Decision Number of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"K. Ehsani, M. Dalirrooyfard, R. Sherkati, S. Akbari, S. Davodpoor","submitted_at":"2014-02-01T22:51:09Z","abstract_excerpt":"Let $G$ be a graph. A good function is a function $f:V(G)\\rightarrow \\{-1,1\\}$, satisfying $f(N(v))\\geq 1$, for each $v\\in V(G)$, where $ N(v)=\\{u\\in V(G)\\, |\\, uv\\in E(G) \\} $ and $f(S) = \\sum_{u\\in S} f(u)$ for every $S \\subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ \\gamma(G) \\leq \\frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\\rightarrow \\{-1,1\\}$ is called a nice function, if $f(N[v])\\le1$, for each $v\\in V(G)$, where $ N[v]=\\{v\\} \\cup N(v) $. Define $\\overline{\\beta}(G)=max\\{f(V(G))\\}$, where $f$ is a nice function for $G$. We show th"},"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":"1402.0134","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-02-01T22:51:09Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"09cd7a66fc662b699a643b471fa88efa7ecb968482be822bbe81793a2f447de9","abstract_canon_sha256":"4486a3a6d5dd765cab6fefea0ffe71da54571b20edbafd3ead6a34b13dacbd35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:00.580828Z","signature_b64":"yRjBFPa/z0ulgwDh3wJtdV6/3HfkbxR3m1GOs3BknMYFCjiE+OeO+wq7Z+OYI/1KYWk3HAk9FQ1W8uy7nIBUDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e720b5b69bb9f2b817696fa371da6926c665f763d7cf51c713709fbf769275d","last_reissued_at":"2026-05-18T02:50:00.580136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:00.580136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Decision Number of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"K. Ehsani, M. Dalirrooyfard, R. Sherkati, S. Akbari, S. Davodpoor","submitted_at":"2014-02-01T22:51:09Z","abstract_excerpt":"Let $G$ be a graph. A good function is a function $f:V(G)\\rightarrow \\{-1,1\\}$, satisfying $f(N(v))\\geq 1$, for each $v\\in V(G)$, where $ N(v)=\\{u\\in V(G)\\, |\\, uv\\in E(G) \\} $ and $f(S) = \\sum_{u\\in S} f(u)$ for every $S \\subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ \\gamma(G) \\leq \\frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\\rightarrow \\{-1,1\\}$ is called a nice function, if $f(N[v])\\le1$, for each $v\\in V(G)$, where $ N[v]=\\{v\\} \\cup N(v) $. Define $\\overline{\\beta}(G)=max\\{f(V(G))\\}$, where $f$ is a nice function for $G$. 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