{"paper":{"title":"The minimal size of a graph with generalized connectivity $\\kappa_3 = 2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shasha Li, Xueliang Li, Yongtang Shi","submitted_at":"2011-01-20T02:30:24Z","abstract_excerpt":"Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\\leq k\\leq n$. For a set $S$ of $k$ vertices of $G$, let $\\kappa (S)$ denote the maximum number $\\ell$ of edge-disjoint trees $T_1,T_2,...,T_\\ell$ in $G$ such that $V(T_i)\\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\\leq i,j\\leq \\ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $\\kappa_k(G)$, of $G$ is defined by $\\kappa_k(G)=$min$\\{\\kappa(S)\\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $\\kappa_2(G)=\\kappa(G)$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3811","kind":"arxiv","version":3},"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"}