{"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? In this paper, we prove that there exists a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with odd $\\Delta(G)\\geq 9$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2415","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"}