{"paper":{"title":"Toughness of recursively partitionable graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brandon Du Preez, Calum Buchanan, K. E. Perry, Puck Rombach","submitted_at":"2022-10-19T14:31:59Z","abstract_excerpt":"A simple graph $G=(V,E)$ on $n$ vertices is said to be recursively partitionable (RP) if $G \\simeq K_1$, or if $G$ is connected and satisfies the following recursive property: for every integer partition $a_1, a_2, \\dots, a_k$ of $n$, there is a partition $\\{A_1, A_2, \\dots, A_k\\}$ of $V$ such that each $|A_i|=a_i$, and each induced subgraph $G[A_i]$ is RP ($1\\leq i \\leq k$). We show that if $S$ is a vertex cut of an RP graph $G$ with $|S|\\geq 2$, then $G-S$ has at most $3|S|-1$ components. Moreover, this bound is sharp for $|S|=3$. We present two methods for constructing new RP graphs from ol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.10590","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.10590/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}