{"paper":{"title":"Saturation numbers for joins of graphs and characterization of extremal graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinying Hua, Yuejian Peng","submitted_at":"2026-06-20T12:20:03Z","abstract_excerpt":"A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\\mathrm{sat}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Saturation number for the join of a vertex and a graph $F$, denoted by $K_1\\vee F$, has received considerable attention. Cameron and Puleo \\cite{Ca} showed that $\\mathrm{sat}(n,K_1 \\vee F)\\le n-1+\\mathrm{sat}(n-1, F)$ for all $n > |V(F)|$. A natural question is to ask when the above equality holds. Existing results for $\\mathrm{sat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22011","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/2606.22011/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"}