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Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one chord and $C_{3}^{*}$ be the graph consisting of a triangle with a pendant edge. In this paper we prove that a nontrivial uniquely $C_{4}^{+}$-saturated graph $G$ has girth $3$ or $4$. Further, $G$ has girth $4$ if and only if it is a strongly regular graph with special parameters. For $n>18k^{2}-24k+10$ with $k\\geq2$, there are no uniquely $C_{4}^{+}$-saturate"},"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":"2412.17962","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-12-23T20:19:02Z","cross_cats_sorted":[],"title_canon_sha256":"49e47639ce2c33dfb31b02e53d28ce8ad7e4d0d559d878cbae3186effa994634","abstract_canon_sha256":"978ee8d4de1f02e4d3c39b988f5ffc1c2cfd8d58026ba5bdfa71a1e8a267b49b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:53:31.444041Z","signature_b64":"KwWfMdimkb+dGbupGlIRVMPsKqiW75WkE/bcxSVyPMXcLxF9tuY2IB9AdFfijQxseOlBnmSKFGan4BvNmJT4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b66e7243813b8dfc4900c06b5102839ef350a95c0146be92b7d3c68b27360cf","last_reissued_at":"2026-07-05T09:53:31.443627Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:53:31.443627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniquely $C_{4}^{+}$-saturated graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D\\'aniel Gerbner, Kexiang Xu, Wenzhong Liu, Yuying Li","submitted_at":"2024-12-23T20:19:02Z","abstract_excerpt":"A graph $G$ is uniquely $H$-saturated if it contains no copy of a graph $H$ as a subgraph, but adding any new edge into $G$ creates exactly one copy of $H$. Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one chord and $C_{3}^{*}$ be the graph consisting of a triangle with a pendant edge. In this paper we prove that a nontrivial uniquely $C_{4}^{+}$-saturated graph $G$ has girth $3$ or $4$. Further, $G$ has girth $4$ if and only if it is a strongly regular graph with special parameters. 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