{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VJM4I32XQN7Y3LS75NR4N52ZB2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5169679509ec66376b912bea27b8d7d954587ee0990e63779cbe1e0cf2104ba1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-03T12:46:11Z","title_canon_sha256":"b364612377e8c33763cdce0a8f424e42fb693ffa4757e2f563d2305f4bc71ba0"},"schema_version":"1.0","source":{"id":"2606.04824","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04824","created_at":"2026-06-04T01:09:31Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04824v1","created_at":"2026-06-04T01:09:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04824","created_at":"2026-06-04T01:09:31Z"},{"alias_kind":"pith_short_12","alias_value":"VJM4I32XQN7Y","created_at":"2026-06-04T01:09:31Z"},{"alias_kind":"pith_short_16","alias_value":"VJM4I32XQN7Y3LS7","created_at":"2026-06-04T01:09:31Z"},{"alias_kind":"pith_short_8","alias_value":"VJM4I32X","created_at":"2026-06-04T01:09:31Z"}],"graph_snapshots":[{"event_id":"sha256:5fd341f8000bf97384afb56fa09846a171a7bbf852749021ac79a5ddc2c2aa1d","target":"graph","created_at":"2026-06-04T01:09:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.04824/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Maker Breaker Domination Game is a two player game played on a graph $G$ in which the players take turns to claim a vertex from the graph. The aim of the Dominator is to claim the vertices of a dominating set, and the aim of the Staller is to prevent this. In this paper, we consider the following problem: for a given integer $d$, what is the size of the smallest (with respect to the number of vertices) graph with minimum degree $d$ such that the Dominator loses going first? We write $\\beta(d)$ to denote the answer to this question. We determine the precise value of $\\beta(d)$ for $d\\leq 3$","authors_text":"Georg Grasegger, Jakob F\\\"uhrer, Oliver Roche-Newton, Paul Hametner","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-03T12:46:11Z","title":"The minimum degree question for the Maker Breaker Domination Game"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04824","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4fd8657642d9651efcdd5c35e68587b1947d29840ddbf46038e3b4389806c3e2","target":"record","created_at":"2026-06-04T01:09:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5169679509ec66376b912bea27b8d7d954587ee0990e63779cbe1e0cf2104ba1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-03T12:46:11Z","title_canon_sha256":"b364612377e8c33763cdce0a8f424e42fb693ffa4757e2f563d2305f4bc71ba0"},"schema_version":"1.0","source":{"id":"2606.04824","kind":"arxiv","version":1}},"canonical_sha256":"aa59c46f57837f8dae5feb63c6f7590ea9ebf570074fbc884314227d0a3b21fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa59c46f57837f8dae5feb63c6f7590ea9ebf570074fbc884314227d0a3b21fe","first_computed_at":"2026-06-04T01:09:31.324747Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:31.324747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bvxx6LY2KLlKsWE8cVBF3f2Yw+k+41sVB0eGA0QMZ5+02ltr7UWrj2VR50qHskXWCKmaNk0yUoAp6RB3VUs6Cg==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:31.325410Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04824","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4fd8657642d9651efcdd5c35e68587b1947d29840ddbf46038e3b4389806c3e2","sha256:5fd341f8000bf97384afb56fa09846a171a7bbf852749021ac79a5ddc2c2aa1d"],"state_sha256":"a6e9ec0f0cb35cf623475b0233c955b108071f91d3017e8c4d6c25af427abf9c"}