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If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number ($\\gamma_{\\rm tcg}(G)$) $\\gamma_{\\rm cg}(G)$ of $G$. We show that $\\gamma_{\\rm tcg}(G)\\in \\{\\gamma_{\\rm cg}(G), \\gamma_{\\rm cg}(G) + 1, \\gamma_{\\rm cg}(G) + 2\\}$, a"},"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":"2010.04907","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-10T05:40:11Z","cross_cats_sorted":[],"title_canon_sha256":"ea3d4ffcbce309d6965cbca174c47d60f787959097a2db566d97fed17ce49818","abstract_canon_sha256":"f789594a7ddba3acb0449d8ca5080674c9cb8c355d143e7e5037b8883d79b1d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:42:00.602543Z","signature_b64":"6ptC4KzN1FCo/84OCnk5cLFJCD+/wz5fZamDXhT3zee5Vql01Xj5PMelfblGPIsljQSHgjNHVoExAuoKvF4kBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65ef86a174204f0c01f2ee40a198adca2c7856189def73a3ce73661eb542db9f","last_reissued_at":"2026-07-05T01:42:00.602113Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:42:00.602113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Total connected domination game","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Csilla Bujt\\'as, Michael A. 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