{"paper":{"title":"Stability of nontrivial graph pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaomeng Wang, Xing Gao","submitted_at":"2026-06-01T12:59:54Z","abstract_excerpt":"A graph pair $(\\Gamma, \\Sigma)$ is called stable if every automorphism of the direct product $\\Gamma\\times\\Sigma$ is induced componentwise by automorphisms of $\\Gamma$ and $\\Sigma$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $\\Gamma$ and $\\Sigma$ are coprime with respect to the direct product if there is no graph $\\Delta$ of order greater than $1$ such that $\\Gamma\\cong\\Gamma'\\times\\Delta$ and $\\Sigma\\cong\\Sigma'\\times\\Delta$ for some graphs $\\Gamma'$ and $\\Sigma'$. A graph pair $(\\Gamma,\\Sigma)$ is nontrivial if $\\Gamma$ and $\\Sigma$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02203","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/2606.02203/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"}