{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:CMH4NJVXSF46NHC7GPG5I4POIT","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":"17c157810cbede749d245875bc63e1452c07f177dea0c281abb2e8051177c03c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-15T10:59:18Z","title_canon_sha256":"77d71c835f41f079ddda892f1228411795b87ac863e08bd1f6d6cb2483236850"},"schema_version":"1.0","source":{"id":"2507.11194","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.11194","created_at":"2026-07-05T11:37:34Z"},{"alias_kind":"arxiv_version","alias_value":"2507.11194v1","created_at":"2026-07-05T11:37:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.11194","created_at":"2026-07-05T11:37:34Z"},{"alias_kind":"pith_short_12","alias_value":"CMH4NJVXSF46","created_at":"2026-07-05T11:37:34Z"},{"alias_kind":"pith_short_16","alias_value":"CMH4NJVXSF46NHC7","created_at":"2026-07-05T11:37:34Z"},{"alias_kind":"pith_short_8","alias_value":"CMH4NJVX","created_at":"2026-07-05T11:37:34Z"}],"graph_snapshots":[{"event_id":"sha256:58f258d35c9a19677221935efed66f0bd8d46e5ad7e3c1b1ac75048a47553155","target":"graph","created_at":"2026-07-05T11:37:34Z","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/2507.11194/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A connected forcing set of a graph is a zero forcing set that induces a connected subgraph. In this paper, we introduce and study CF-dense graphs -- graphs in which every vertex belongs to some minimum connected forcing set. We identify several CF-dense graph families and investigate the relationships between CF-density and analogous notions in zero forcing and total forcing. We also characterize CF-dense trees and give a formula for the number of distinct connected forcing sets in trees. Finally, we analyze when CF-density is preserved under graph operations such as Cartesian products, joins,","authors_text":"Boris Brimkov, Houston Schuerger, Randy Davila","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-15T10:59:18Z","title":"Connected forcing density and related problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11194","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":"sha256:7e530a75561d2ab7abd5fd915651acd19ecb106f55f8f0782dcca6de60039224","target":"record","created_at":"2026-07-05T11:37:34Z","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":"17c157810cbede749d245875bc63e1452c07f177dea0c281abb2e8051177c03c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-15T10:59:18Z","title_canon_sha256":"77d71c835f41f079ddda892f1228411795b87ac863e08bd1f6d6cb2483236850"},"schema_version":"1.0","source":{"id":"2507.11194","kind":"arxiv","version":1}},"canonical_sha256":"130fc6a6b79179e69c5f33cdd471ee44cdbdd1b94077a768d01969680e043e07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"130fc6a6b79179e69c5f33cdd471ee44cdbdd1b94077a768d01969680e043e07","first_computed_at":"2026-07-05T11:37:34.090743Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:37:34.090743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AF00Wtj1ye/BqpdT8SEG6Rz1nk38mrrANBnvQQfRoB5KiXe0xfDcBnLbVhnQSUZPYUYF9GOpcCEnoniVs1CPAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:37:34.091234Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.11194","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e530a75561d2ab7abd5fd915651acd19ecb106f55f8f0782dcca6de60039224","sha256:58f258d35c9a19677221935efed66f0bd8d46e5ad7e3c1b1ac75048a47553155"],"state_sha256":"9f3477f29da087e8c825c05ff6e24b330fc5cdb4821e6601a67f8e3bc1e93e43"}