{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JQJ4VHCAKYKJZL426PP3RDRQSO","short_pith_number":"pith:JQJ4VHCA","canonical_record":{"source":{"id":"1507.06141","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-22T11:56:20Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"802fc06de4bd780164fe1d5819dd19ceb9ab1ceb4812b7ba39806df18e12f0af","abstract_canon_sha256":"9c05ff1c076b2bb7df025c3944d5984c48f1de2b18142b6d5e6a2a10710985b3"},"schema_version":"1.0"},"canonical_sha256":"4c13ca9c4056149caf9af3dfb88e3093a57878527cf9addd70c9e5f30a1de351","source":{"kind":"arxiv","id":"1507.06141","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06141","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06141v2","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06141","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"pith_short_12","alias_value":"JQJ4VHCAKYKJ","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JQJ4VHCAKYKJZL42","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JQJ4VHCA","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JQJ4VHCAKYKJZL426PP3RDRQSO","target":"record","payload":{"canonical_record":{"source":{"id":"1507.06141","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-22T11:56:20Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"802fc06de4bd780164fe1d5819dd19ceb9ab1ceb4812b7ba39806df18e12f0af","abstract_canon_sha256":"9c05ff1c076b2bb7df025c3944d5984c48f1de2b18142b6d5e6a2a10710985b3"},"schema_version":"1.0"},"canonical_sha256":"4c13ca9c4056149caf9af3dfb88e3093a57878527cf9addd70c9e5f30a1de351","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:14.121307Z","signature_b64":"lkoqJUPTiauDAIQsh5IdrY6vtlL8ca65afoHAXu7mlBKbd9IqwiQivyH1R7cee5K/iOamM1eaD5F3KdgMNbXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c13ca9c4056149caf9af3dfb88e3093a57878527cf9addd70c9e5f30a1de351","last_reissued_at":"2026-05-18T00:31:14.120694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:14.120694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.06141","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:31:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aMLiaHPqSlmDtq8sRhRbEeR2SXTw4uZCi8x9NaYValT3zfmAGRyF44x4JZUa3YEur58t1CKkoEI21XDlefwxBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:43:18.015520Z"},"content_sha256":"9501f233150d89017ab6bf5c24b77151b16e3a147d0ca222e38102a9e899e4de","schema_version":"1.0","event_id":"sha256:9501f233150d89017ab6bf5c24b77151b16e3a147d0ca222e38102a9e899e4de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JQJ4VHCAKYKJZL426PP3RDRQSO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fixed points in conjunctive networks and maximal independent sets in graph contractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Adrien Richard, Julio Aracena, Lilian Salinas","submitted_at":"2015-07-22T11:56:20Z","abstract_excerpt":"Given a graph $G$, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with $G$ as interaction graph. We prove that if $G$ has no induced $C_4$, then this quantity equals both the number of maximal independent sets in $G$ and the maximum number of maximal independent sets among all the graphs obtained from $G$ by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if $G$ has a unique induced $C_4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06141","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:31:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nVFCL9lZhhyowWP1f5BxjWAdTBvz2jxzRIk8sQGKWgDoqaSbjhFFtuuxpkVnO0ncu1XBNm30Isn33hJu0KbCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:43:18.015886Z"},"content_sha256":"e8a2cd5a9437bee9388fd393ad5caea38b1a289971c728cda98ba65e4847d8e1","schema_version":"1.0","event_id":"sha256:e8a2cd5a9437bee9388fd393ad5caea38b1a289971c728cda98ba65e4847d8e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/bundle.json","state_url":"https://pith.science/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T14:43:18Z","links":{"resolver":"https://pith.science/pith/JQJ4VHCAKYKJZL426PP3RDRQSO","bundle":"https://pith.science/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/bundle.json","state":"https://pith.science/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JQJ4VHCAKYKJZL426PP3RDRQSO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JQJ4VHCAKYKJZL426PP3RDRQSO","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":"9c05ff1c076b2bb7df025c3944d5984c48f1de2b18142b6d5e6a2a10710985b3","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-22T11:56:20Z","title_canon_sha256":"802fc06de4bd780164fe1d5819dd19ceb9ab1ceb4812b7ba39806df18e12f0af"},"schema_version":"1.0","source":{"id":"1507.06141","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.06141","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"arxiv_version","alias_value":"1507.06141v2","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06141","created_at":"2026-05-18T00:31:14Z"},{"alias_kind":"pith_short_12","alias_value":"JQJ4VHCAKYKJ","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JQJ4VHCAKYKJZL42","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JQJ4VHCA","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:e8a2cd5a9437bee9388fd393ad5caea38b1a289971c728cda98ba65e4847d8e1","target":"graph","created_at":"2026-05-18T00:31:14Z","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"},"paper":{"abstract_excerpt":"Given a graph $G$, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with $G$ as interaction graph. We prove that if $G$ has no induced $C_4$, then this quantity equals both the number of maximal independent sets in $G$ and the maximum number of maximal independent sets among all the graphs obtained from $G$ by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if $G$ has a unique induced $C_4$.","authors_text":"Adrien Richard, Julio Aracena, Lilian Salinas","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-22T11:56:20Z","title":"Fixed points in conjunctive networks and maximal independent sets in graph contractions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06141","kind":"arxiv","version":2},"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":"sha256:9501f233150d89017ab6bf5c24b77151b16e3a147d0ca222e38102a9e899e4de","target":"record","created_at":"2026-05-18T00:31:14Z","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":"9c05ff1c076b2bb7df025c3944d5984c48f1de2b18142b6d5e6a2a10710985b3","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-22T11:56:20Z","title_canon_sha256":"802fc06de4bd780164fe1d5819dd19ceb9ab1ceb4812b7ba39806df18e12f0af"},"schema_version":"1.0","source":{"id":"1507.06141","kind":"arxiv","version":2}},"canonical_sha256":"4c13ca9c4056149caf9af3dfb88e3093a57878527cf9addd70c9e5f30a1de351","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c13ca9c4056149caf9af3dfb88e3093a57878527cf9addd70c9e5f30a1de351","first_computed_at":"2026-05-18T00:31:14.120694Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:14.120694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lkoqJUPTiauDAIQsh5IdrY6vtlL8ca65afoHAXu7mlBKbd9IqwiQivyH1R7cee5K/iOamM1eaD5F3KdgMNbXBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:14.121307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.06141","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9501f233150d89017ab6bf5c24b77151b16e3a147d0ca222e38102a9e899e4de","sha256:e8a2cd5a9437bee9388fd393ad5caea38b1a289971c728cda98ba65e4847d8e1"],"state_sha256":"5e8b8e37e334d75ca0eb58eeae05bc6204f06faee00c55170d29f14bb3767ec4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JbmXFfyvsjDHhIx5bNrJEfvTL04V+uMOizalyYAs7IUNk7WEwoYUL84Cwe4queCQW+w4V0v447kT5aDW6U8bBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T14:43:18.017808Z","bundle_sha256":"03e5c031b82400d564929518e132e74526ab6c6dcd670e1adcf7ca17f8fdbaa6"}}