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The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with $\\g_p(G')<\\g_p(G)$. Recently, it was proved by Lu et al. that $r_p(T)\\leq p+1$ for a tree $T$ and $p\\geq 2$. In this paper, we characterize all trees attaining this upper bound for $p\\geq 3$."},"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":"1211.5742","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-25T08:35:46Z","cross_cats_sorted":[],"title_canon_sha256":"c27a7cc9e6b00846bdf08526d4a7f322bbb15103170f5970afa97ca2f7fcfff4","abstract_canon_sha256":"b091801f1bac67de6354cb1e25f2688baf839f046b0dd90f700dacd4ccd25f24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:02.072838Z","signature_b64":"GvmeMxX4upR6dXDsT8FPbIJ79/njsiUFnFWI493tHIRKMJNMYPXEP8perejJ/+d4pmlOkF66rpZaztcPJKtJCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2684baa2846184e2240def513348efd03045d721348a502c967e19766a19f71","last_reissued_at":"2026-05-18T03:40:02.072365Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:02.072365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Trees with Maximum p-Reinforcement Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun-Ming Xu, You Lu","submitted_at":"2012-11-25T08:35:46Z","abstract_excerpt":"Let $G=(V,E)$ be a graph and $p$ a positive integer. 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