{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:A2UKOV6BBW5KB4XES7RMJ3PW5Z","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":"5b5d86a43cb7cdc6651b8deeeaf04625f352d47ef95dac8fc2d39aed7853c1fd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2017-02-17T17:20:21Z","title_canon_sha256":"ac8e23a979af165bfd94daa7bd561b23e912c1d94541efb1981521c7ff2f2215"},"schema_version":"1.0","source":{"id":"1702.05447","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.05447","created_at":"2026-05-18T00:25:34Z"},{"alias_kind":"arxiv_version","alias_value":"1702.05447v2","created_at":"2026-05-18T00:25:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.05447","created_at":"2026-05-18T00:25:34Z"},{"alias_kind":"pith_short_12","alias_value":"A2UKOV6BBW5K","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"A2UKOV6BBW5KB4XE","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"A2UKOV6B","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:058ffbc68cbdc5a6d734d41256e9161af95f0f325bfa545f4d45508eaaad6719","target":"graph","created_at":"2026-05-18T00:25:34Z","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"},"paper":{"abstract_excerpt":"We consider the $\\#\\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\\#\\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective h","authors_text":"Holger Dell, Marc Roth, Radu Curticapean","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2017-02-17T17:20:21Z","title":"Counting edge-injective homomorphisms and matchings on restricted graph classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05447","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:e318f0b0f11926baf9150c54e9f915ec237e2cca60644590b937f12b7a269b50","target":"record","created_at":"2026-05-18T00:25:34Z","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":"5b5d86a43cb7cdc6651b8deeeaf04625f352d47ef95dac8fc2d39aed7853c1fd","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2017-02-17T17:20:21Z","title_canon_sha256":"ac8e23a979af165bfd94daa7bd561b23e912c1d94541efb1981521c7ff2f2215"},"schema_version":"1.0","source":{"id":"1702.05447","kind":"arxiv","version":2}},"canonical_sha256":"06a8a757c10dbaa0f2e497e2c4edf6ee76211905ec84f67ad2d7dcf85d1476ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06a8a757c10dbaa0f2e497e2c4edf6ee76211905ec84f67ad2d7dcf85d1476ac","first_computed_at":"2026-05-18T00:25:34.617751Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:25:34.617751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M+z3OvDaP2bEtcpbulqxNBf+GwkYT8NAEngwNiKLsRxj51xjg1Hql/fbVTeb+eMo9z9NtVPQaYpe8AafdO6vDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:25:34.618476Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.05447","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e318f0b0f11926baf9150c54e9f915ec237e2cca60644590b937f12b7a269b50","sha256:058ffbc68cbdc5a6d734d41256e9161af95f0f325bfa545f4d45508eaaad6719"],"state_sha256":"86187e6bc269b68e81a334bfd91fb6b1e6d58c965486bdf14bfa5b919a0e76fa"}