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One of the problems about it is to determine the largest number of edges $f(n;\\bar{\\kappa}\\leq \\ell)$ for graphs of order $n$ that have local connectivity at most $\\ell$. We consider a generalization of the above concept and problem. For $S\\subseteq V(G)$ and $|S|\\geq 2$, the \\emph{generalized local connectivity} $\\kappa(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\\bar{\\kappa}_k(G)=max\\{\\kappa(S)|S\\subseteq V(G),|S|=k\\}$ is called the \\emph{maxim"},"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":"1304.3774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-13T04:23:23Z","cross_cats_sorted":[],"title_canon_sha256":"d0a5e30cccfd3b60ab44b6a66f64a1b169ad5fb646daa6d3ab58733301befeb0","abstract_canon_sha256":"08b178d0aa163ffe03eb45ee33186872f548130e81b79e4073b5938ed5a4eb98"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:04.393503Z","signature_b64":"Mno+mSr0cgx85jui4ZVISfhV5szscJndsFvWcmTKIjoSALR6OIu8T5nxiDzE7W7YW8DtQAwFRmYYSb+aEMWgBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7cdac67aef847764a0227d086ef3da19173dedd59ceb1d196910bb5670404f0","last_reissued_at":"2026-05-18T03:28:04.392242Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:04.392242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On extremal graphs with at most $\\ell$ internally disjoint Steiner trees connecting any n-1 vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xueliang Li, Yaping Mao","submitted_at":"2013-04-13T04:23:23Z","abstract_excerpt":"The concept of maximum local connectivity $\\bar {\\kappa}$ of a graph was introduced by Bollob\\'{a}s. 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