{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:L5KJDOJZGVDA27PIGSSHM24DFN","short_pith_number":"pith:L5KJDOJZ","canonical_record":{"source":{"id":"1501.07539","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-29T18:35:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"89a45066c926f461e07404ee5572ae407fedb011d498c245df580c60fcd6e18c","abstract_canon_sha256":"cd5765b829f0c5855e899ed285e79dce545b6e374972fd98137369ca4e8bc2e6"},"schema_version":"1.0"},"canonical_sha256":"5f5491b93935460d7de834a4766b832b5906f6afba1eba8aee749a11ee948d6d","source":{"kind":"arxiv","id":"1501.07539","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07539","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07539v4","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07539","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"L5KJDOJZGVDA","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"L5KJDOJZGVDA27PI","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"L5KJDOJZ","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:L5KJDOJZGVDA27PIGSSHM24DFN","target":"record","payload":{"canonical_record":{"source":{"id":"1501.07539","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-29T18:35:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"89a45066c926f461e07404ee5572ae407fedb011d498c245df580c60fcd6e18c","abstract_canon_sha256":"cd5765b829f0c5855e899ed285e79dce545b6e374972fd98137369ca4e8bc2e6"},"schema_version":"1.0"},"canonical_sha256":"5f5491b93935460d7de834a4766b832b5906f6afba1eba8aee749a11ee948d6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:45.366449Z","signature_b64":"hFXBVfUHIuBrQPolPng54h2C8PkNbjNdfELb0TZGakQ33ZUEv05xZQ/BocA6ezLAoWYh48RkfnAlyF+HODgBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f5491b93935460d7de834a4766b832b5906f6afba1eba8aee749a11ee948d6d","last_reissued_at":"2026-05-18T01:34:45.365778Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:45.365778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.07539","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IRgP0VpPP0IRYg6/5GyLUynaiLdkQv+2g36M12Jhx8CsMmFVDSzz0PsOjfO9mJbJV5fY4GgC68AsyE4NB7ePBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:21:56.908576Z"},"content_sha256":"799ccd980e04eb0d2e3861b8dfefcd885e770b45db91ce17285a881350921dd1","schema_version":"1.0","event_id":"sha256:799ccd980e04eb0d2e3861b8dfefcd885e770b45db91ce17285a881350921dd1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:L5KJDOJZGVDA27PIGSSHM24DFN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting Homomorphisms to Square-Free Graphs, Modulo 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CC","authors_text":"Andreas G\\\"obel, David Richerby, Leslie Ann Goldberg","submitted_at":"2015-01-29T18:35:52Z","abstract_excerpt":"We study the problem HomsTo$H$ of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph $H$. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any $H$ that contains no 4-cycles, HomsTo$H$ is either in polynomial time or is $\\oplus P$-complete. This confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth-2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07539","kind":"arxiv","version":4},"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":"2026-05-18T01:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NJmK+wh3UhIRkBNUOvJD/UPbgDY432Qt1yCMCvfxVTm3gADrd9TAqQwL2+eVqGdFRcdgEoIdoZvPoaa9UvrMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:21:56.909297Z"},"content_sha256":"e5f5b0a6fa57a0be48383e60d54d38cf8d9557ce3489673920a60d1564c8c986","schema_version":"1.0","event_id":"sha256:e5f5b0a6fa57a0be48383e60d54d38cf8d9557ce3489673920a60d1564c8c986"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L5KJDOJZGVDA27PIGSSHM24DFN/bundle.json","state_url":"https://pith.science/pith/L5KJDOJZGVDA27PIGSSHM24DFN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L5KJDOJZGVDA27PIGSSHM24DFN/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":"2026-05-31T11:21:56Z","links":{"resolver":"https://pith.science/pith/L5KJDOJZGVDA27PIGSSHM24DFN","bundle":"https://pith.science/pith/L5KJDOJZGVDA27PIGSSHM24DFN/bundle.json","state":"https://pith.science/pith/L5KJDOJZGVDA27PIGSSHM24DFN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L5KJDOJZGVDA27PIGSSHM24DFN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:L5KJDOJZGVDA27PIGSSHM24DFN","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":"cd5765b829f0c5855e899ed285e79dce545b6e374972fd98137369ca4e8bc2e6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-29T18:35:52Z","title_canon_sha256":"89a45066c926f461e07404ee5572ae407fedb011d498c245df580c60fcd6e18c"},"schema_version":"1.0","source":{"id":"1501.07539","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07539","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07539v4","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07539","created_at":"2026-05-18T01:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"L5KJDOJZGVDA","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"L5KJDOJZGVDA27PI","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"L5KJDOJZ","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:e5f5b0a6fa57a0be48383e60d54d38cf8d9557ce3489673920a60d1564c8c986","target":"graph","created_at":"2026-05-18T01:34:45Z","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":"We study the problem HomsTo$H$ of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph $H$. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any $H$ that contains no 4-cycles, HomsTo$H$ is either in polynomial time or is $\\oplus P$-complete. This confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth-2","authors_text":"Andreas G\\\"obel, David Richerby, Leslie Ann Goldberg","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-29T18:35:52Z","title":"Counting Homomorphisms to Square-Free Graphs, Modulo 2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07539","kind":"arxiv","version":4},"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":"sha256:799ccd980e04eb0d2e3861b8dfefcd885e770b45db91ce17285a881350921dd1","target":"record","created_at":"2026-05-18T01:34:45Z","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":"cd5765b829f0c5855e899ed285e79dce545b6e374972fd98137369ca4e8bc2e6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-29T18:35:52Z","title_canon_sha256":"89a45066c926f461e07404ee5572ae407fedb011d498c245df580c60fcd6e18c"},"schema_version":"1.0","source":{"id":"1501.07539","kind":"arxiv","version":4}},"canonical_sha256":"5f5491b93935460d7de834a4766b832b5906f6afba1eba8aee749a11ee948d6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f5491b93935460d7de834a4766b832b5906f6afba1eba8aee749a11ee948d6d","first_computed_at":"2026-05-18T01:34:45.365778Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:45.365778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hFXBVfUHIuBrQPolPng54h2C8PkNbjNdfELb0TZGakQ33ZUEv05xZQ/BocA6ezLAoWYh48RkfnAlyF+HODgBAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:45.366449Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.07539","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:799ccd980e04eb0d2e3861b8dfefcd885e770b45db91ce17285a881350921dd1","sha256:e5f5b0a6fa57a0be48383e60d54d38cf8d9557ce3489673920a60d1564c8c986"],"state_sha256":"edb88cf80f3f3d74da1bba788d1285594eeb818352acf40ecdf8c2634a6a11d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TAePJgMYEFFqhMLLubCLnltLNt/u37+XoacIgDLQCverE+lN3OjDpdMRJYkq0kn27nrG/igKhfoydzzLNgJ7BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T11:21:56.913527Z","bundle_sha256":"27b763ad072a2d64b9d1bf632039e0ce24e46c524b2477a29abc41be33218ccb"}}