{"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. The $p$-domination number $\\g_p(G)$ is the minimum cardinality of a set $D\\subseteq V$ with $|N_G(x)\\cap D|\\geq p$ for all $x\\in V\\setminus D$. 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 $p\\geq 3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5742","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"}