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The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the {\\em toppling number} of a graph $G$, and is denoted by $\\tg(G)$.\n  By cons"},"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":"1305.1677","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-07T23:12:00Z","cross_cats_sorted":[],"title_canon_sha256":"51c327a9a097afdfe38f8634df0dcb1df70f1ca5127c78b25dda5b02e2e49353","abstract_canon_sha256":"502a3437c8c15c888b602645a3281643ffda74789ab9dc680fde08ed61057e9d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:11.606886Z","signature_b64":"DdRcrqdeWmK5SCb0F8QlY1O3BKZRm2plsNYukF7NlDhDgM+53NbARTHRHLy7IGk/+HnMvbiJMn0N4ueh7LH5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5444846226522938a6e0b3156964caae9530b597fdb265cba629fbab86fc299b","last_reissued_at":"2026-05-18T03:26:11.606108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:11.606108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Toppling numbers of complete and random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. 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