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Mojdeh and Rad \\cite{MR06} proposed an open problem: Does there exist a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with $\\Delta(G)$ odd? In this paper, we prove that there exists a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with odd $\\Delta(G)\\geq 9$."},"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":"1103.2415","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","cross_cats_sorted":[],"title_canon_sha256":"c6ec2c2eb06c03a85a088c856d9580f3438b6432fb5bd6199fbc321e98ca801a","abstract_canon_sha256":"eec4fe4158fdd3b560604918cabf8584a6367f15d873ebce53a55dd59f5cc0ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:55.321169Z","signature_b64":"1gwAl/sGX1rN1Ud59A9yuaLsB+jnF9RR3RqGUoAjwDoj7V+1D84cMPHQ9oQLXfl+EQdAYIw8s4xweHRkkSd9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","last_reissued_at":"2026-05-18T04:26:55.320583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:55.320583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the 3-$\\gamma_t$-Critical Graphs of Order $\\Delta(G)+3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haoli Wang, Lei Wang, Xirong Xu, Yang Yuansheng","submitted_at":"2011-03-12T03:38:26Z","abstract_excerpt":"Let $\\gamma_t(G)$ be the total domination number of graph $G$, a graph $G$ is $k$-total domination vertex critical (or\\ just\\ $k$-$\\gamma_t$-critical) if $\\gamma_t(G)=k$, and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\\gamma_t(G-v)=k-1$. Mojdeh and Rad \\cite{MR06} proposed an open problem: Does there exist a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with $\\Delta(G)$ odd? 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