{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GZAQHS4YMQ6BHNPGG6COI52RLB","short_pith_number":"pith:GZAQHS4Y","canonical_record":{"source":{"id":"2607.04185","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T09:03:54Z","cross_cats_sorted":[],"title_canon_sha256":"e12627f8a0380dd8e861779040ca9c06479df51755f0a79f847b533dba92c48a","abstract_canon_sha256":"36cbdb64a6486da697922d681a76dd898d87e4c14c2ed3885acba1aaf38abca6"},"schema_version":"1.0"},"canonical_sha256":"364103cb98643c13b5e63784e47751584670463fc44ab98b8bee50ad8f54c930","source":{"kind":"arxiv","id":"2607.04185","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04185","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04185v1","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04185","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_12","alias_value":"GZAQHS4YMQ6B","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_16","alias_value":"GZAQHS4YMQ6BHNPG","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_8","alias_value":"GZAQHS4Y","created_at":"2026-07-07T02:19:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GZAQHS4YMQ6BHNPGG6COI52RLB","target":"record","payload":{"canonical_record":{"source":{"id":"2607.04185","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T09:03:54Z","cross_cats_sorted":[],"title_canon_sha256":"e12627f8a0380dd8e861779040ca9c06479df51755f0a79f847b533dba92c48a","abstract_canon_sha256":"36cbdb64a6486da697922d681a76dd898d87e4c14c2ed3885acba1aaf38abca6"},"schema_version":"1.0"},"canonical_sha256":"364103cb98643c13b5e63784e47751584670463fc44ab98b8bee50ad8f54c930","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:01.495471Z","signature_b64":"0Ka1bNS9yXWONkOFnjoiDRVCZgpXtZ6Jj2xgXoL+tFf8+Dl0DlJZ5agdnS/qO6LP3n2011s13s9sCU8fkphLBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"364103cb98643c13b5e63784e47751584670463fc44ab98b8bee50ad8f54c930","last_reissued_at":"2026-07-07T02:19:01.494693Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:01.494693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.04185","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-07T02:19:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bgC6lHOcQNPxY/GyAbVqkS1B2TGRWO4seVcQC82SsXTz0s8hSGTRhzP7H20PET83VqzkKRVZqPFI2RO6PtEoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T10:14:47.148232Z"},"content_sha256":"b11e570d199744843f65c08f422b3295d475f2a55193495275dab0de7e8f59ec","schema_version":"1.0","event_id":"sha256:b11e570d199744843f65c08f422b3295d475f2a55193495275dab0de7e8f59ec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GZAQHS4YMQ6BHNPGG6COI52RLB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Complexity of Weak Saturation for Complete graphs and Balanced Complete Bipartite Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tianying Xie, Yihan Chen","submitted_at":"2026-07-05T09:03:54Z","abstract_excerpt":"For graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is weakly $H$-saturated in $F$ if the edges in $E(F)\\setminus E(G)$ can be added one at a time, each addition creating a new copy of $H$. Recently, Tancer and Tyomkyn proved that, given an $n$-vertex graph $F$, deciding whether $\\mathrm{wsat}(F,K_3)=n-1$ is NP-hard. In this paper, we study the decision version of the weak saturation problem and show that, for every fixed integer $r\\ge 3$, given a graph $F$ and an integer $k$, deciding whether $\\mathrm{wsat}(F,H)\\le k$ is NP-complete when $H\\in\\{K_r,K_{r,r}\\}$. Our approach uses new graph-t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04185","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.04185/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-07T02:19:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ITwVdlLUmyXetPG+xUlOOiYOLlKYAhJ1BSwpmGWR2kqOlNluEKxyZM49r29PhPtCHW1vd6w7C0R2OlChXruaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T10:14:47.148713Z"},"content_sha256":"8ae5bea248dc75f542bc59fafa97125758b1c613922bdd792f320590354f7528","schema_version":"1.0","event_id":"sha256:8ae5bea248dc75f542bc59fafa97125758b1c613922bdd792f320590354f7528"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/bundle.json","state_url":"https://pith.science/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/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":"2026-08-21T10:14:47Z","links":{"resolver":"https://pith.science/pith/GZAQHS4YMQ6BHNPGG6COI52RLB","bundle":"https://pith.science/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/bundle.json","state":"https://pith.science/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GZAQHS4YMQ6BHNPGG6COI52RLB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GZAQHS4YMQ6BHNPGG6COI52RLB","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":"36cbdb64a6486da697922d681a76dd898d87e4c14c2ed3885acba1aaf38abca6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T09:03:54Z","title_canon_sha256":"e12627f8a0380dd8e861779040ca9c06479df51755f0a79f847b533dba92c48a"},"schema_version":"1.0","source":{"id":"2607.04185","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04185","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04185v1","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04185","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_12","alias_value":"GZAQHS4YMQ6B","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_16","alias_value":"GZAQHS4YMQ6BHNPG","created_at":"2026-07-07T02:19:01Z"},{"alias_kind":"pith_short_8","alias_value":"GZAQHS4Y","created_at":"2026-07-07T02:19:01Z"}],"graph_snapshots":[{"event_id":"sha256:8ae5bea248dc75f542bc59fafa97125758b1c613922bdd792f320590354f7528","target":"graph","created_at":"2026-07-07T02:19:01Z","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/2607.04185/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is weakly $H$-saturated in $F$ if the edges in $E(F)\\setminus E(G)$ can be added one at a time, each addition creating a new copy of $H$. Recently, Tancer and Tyomkyn proved that, given an $n$-vertex graph $F$, deciding whether $\\mathrm{wsat}(F,K_3)=n-1$ is NP-hard. In this paper, we study the decision version of the weak saturation problem and show that, for every fixed integer $r\\ge 3$, given a graph $F$ and an integer $k$, deciding whether $\\mathrm{wsat}(F,H)\\le k$ is NP-complete when $H\\in\\{K_r,K_{r,r}\\}$. Our approach uses new graph-t","authors_text":"Tianying Xie, Yihan Chen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T09:03:54Z","title":"The Complexity of Weak Saturation for Complete graphs and Balanced Complete Bipartite Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04185","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:b11e570d199744843f65c08f422b3295d475f2a55193495275dab0de7e8f59ec","target":"record","created_at":"2026-07-07T02:19:01Z","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":"36cbdb64a6486da697922d681a76dd898d87e4c14c2ed3885acba1aaf38abca6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-05T09:03:54Z","title_canon_sha256":"e12627f8a0380dd8e861779040ca9c06479df51755f0a79f847b533dba92c48a"},"schema_version":"1.0","source":{"id":"2607.04185","kind":"arxiv","version":1}},"canonical_sha256":"364103cb98643c13b5e63784e47751584670463fc44ab98b8bee50ad8f54c930","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"364103cb98643c13b5e63784e47751584670463fc44ab98b8bee50ad8f54c930","first_computed_at":"2026-07-07T02:19:01.494693Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:19:01.494693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0Ka1bNS9yXWONkOFnjoiDRVCZgpXtZ6Jj2xgXoL+tFf8+Dl0DlJZ5agdnS/qO6LP3n2011s13s9sCU8fkphLBA==","signature_status":"signed_v1","signed_at":"2026-07-07T02:19:01.495471Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.04185","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b11e570d199744843f65c08f422b3295d475f2a55193495275dab0de7e8f59ec","sha256:8ae5bea248dc75f542bc59fafa97125758b1c613922bdd792f320590354f7528"],"state_sha256":"8df063af04297cf9b6bc5ece584e7ab644e89940caefda8df7b68f7f16870168"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vORtcTGL8bRsKnaCETfX5AkEr9TR6SGaIfQItha2U6w7urwytkdTdvSVpdxzBfRLNOhdg5MoESDVFa13Fs3XCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T10:14:47.153767Z","bundle_sha256":"f3a6f43397e32f4dbf985b6577baa22d2f1090f6fc41037829510d061d52456d"}}