{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:RTTLAZQGED2YPKFJQVQXPEJ66P","short_pith_number":"pith:RTTLAZQG","canonical_record":{"source":{"id":"2606.00719","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-30T13:05:05Z","cross_cats_sorted":[],"title_canon_sha256":"2ed3761f90cc3437fc82e2aee4bedc8add5fe3cb1088eb04167a92b7ca7ed9b7","abstract_canon_sha256":"7fb993b74e09f1f74fa7f1ca74c529e1c0600b92aabcb13597f2b67e6177f096"},"schema_version":"1.0"},"canonical_sha256":"8ce6b0660620f587a8a9856177913ef3f9d550fa8fd0936404ab7ee8d760b86e","source":{"kind":"arxiv","id":"2606.00719","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00719","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00719v1","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00719","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_12","alias_value":"RTTLAZQGED2Y","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_16","alias_value":"RTTLAZQGED2YPKFJ","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_8","alias_value":"RTTLAZQG","created_at":"2026-06-02T01:04:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:RTTLAZQGED2YPKFJQVQXPEJ66P","target":"record","payload":{"canonical_record":{"source":{"id":"2606.00719","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-30T13:05:05Z","cross_cats_sorted":[],"title_canon_sha256":"2ed3761f90cc3437fc82e2aee4bedc8add5fe3cb1088eb04167a92b7ca7ed9b7","abstract_canon_sha256":"7fb993b74e09f1f74fa7f1ca74c529e1c0600b92aabcb13597f2b67e6177f096"},"schema_version":"1.0"},"canonical_sha256":"8ce6b0660620f587a8a9856177913ef3f9d550fa8fd0936404ab7ee8d760b86e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:04:03.697342Z","signature_b64":"fX4bgde3mBzC69MGSclrcW7M5yvGp4U82f1Y4EvbuSqqkq9ew7MZSYCYMFO78JOXFyaLBAYNLu0JXQWwc1+nCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ce6b0660620f587a8a9856177913ef3f9d550fa8fd0936404ab7ee8d760b86e","last_reissued_at":"2026-06-02T01:04:03.696936Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:04:03.696936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.00719","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":"2026-06-02T01:04:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q/R60pMdbAgTD4Bd6mYFFqL0vlTLk7ayAEvgiaYPtrHICE1BxgxB3g1E1FVHvHwgDR8XLNej4VQFlhYmZQfiBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:58:21.221790Z"},"content_sha256":"9bec938d34476f860de91a3087cfcc1601f502d58ae39c26b6ba3dcd9aeb9a55","schema_version":"1.0","event_id":"sha256:9bec938d34476f860de91a3087cfcc1601f502d58ae39c26b6ba3dcd9aeb9a55"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:RTTLAZQGED2YPKFJQVQXPEJ66P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Characterization of the structure of $k$-edge-maximal graphs","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong-Jian Lai, Jian Lu, Zheng-Jiang Xia, Zhen-Mu Hong","submitted_at":"2026-05-30T13:05:05Z","abstract_excerpt":"Let $\\kappa^{\\prime}(G)$ be the edge-connectivity of the graph $G$. The \\textit{strength} of $G$, denoted by $\\overline{\\kappa}^{\\prime}(G)$, is the maximum edge-connectivity of its subgraphs. A simple graph $G$ is called $k$-\\textit{edge-maximal} if $\\overline{\\kappa}^{\\prime}(G) \\leq k$ but for any edge $e$ not in $G$, $\\overline{\\kappa}^{\\prime}(G+e) \\geq k+1$. In this paper, we propose the concepts of kernel and closure of a graph and discuss the properties of closure. Utilizing these properties, we present the necessary and sufficient condition for a graph to be $k$-edge-maximal, which re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00719","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/2606.00719/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":"2026-06-02T01:04:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hJhlnqQWq9qsWsepS1/Z8YL1H8EYUvoV9Y9dvzTkhgpiZ6BPbSMLuiCx2x5svKUd4dHMeymhV2jvxtYw9Y5YAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:58:21.222162Z"},"content_sha256":"0cb5aa7ed70e84316efa3a664660a7046934b7b0bea03e582de6e71ae9d1f87a","schema_version":"1.0","event_id":"sha256:0cb5aa7ed70e84316efa3a664660a7046934b7b0bea03e582de6e71ae9d1f87a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/bundle.json","state_url":"https://pith.science/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/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":"2026-06-02T19:58:21Z","links":{"resolver":"https://pith.science/pith/RTTLAZQGED2YPKFJQVQXPEJ66P","bundle":"https://pith.science/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/bundle.json","state":"https://pith.science/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RTTLAZQGED2YPKFJQVQXPEJ66P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RTTLAZQGED2YPKFJQVQXPEJ66P","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":"7fb993b74e09f1f74fa7f1ca74c529e1c0600b92aabcb13597f2b67e6177f096","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-30T13:05:05Z","title_canon_sha256":"2ed3761f90cc3437fc82e2aee4bedc8add5fe3cb1088eb04167a92b7ca7ed9b7"},"schema_version":"1.0","source":{"id":"2606.00719","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.00719","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"arxiv_version","alias_value":"2606.00719v1","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00719","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_12","alias_value":"RTTLAZQGED2Y","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_16","alias_value":"RTTLAZQGED2YPKFJ","created_at":"2026-06-02T01:04:03Z"},{"alias_kind":"pith_short_8","alias_value":"RTTLAZQG","created_at":"2026-06-02T01:04:03Z"}],"graph_snapshots":[{"event_id":"sha256:0cb5aa7ed70e84316efa3a664660a7046934b7b0bea03e582de6e71ae9d1f87a","target":"graph","created_at":"2026-06-02T01:04:03Z","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/2606.00719/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\kappa^{\\prime}(G)$ be the edge-connectivity of the graph $G$. The \\textit{strength} of $G$, denoted by $\\overline{\\kappa}^{\\prime}(G)$, is the maximum edge-connectivity of its subgraphs. A simple graph $G$ is called $k$-\\textit{edge-maximal} if $\\overline{\\kappa}^{\\prime}(G) \\leq k$ but for any edge $e$ not in $G$, $\\overline{\\kappa}^{\\prime}(G+e) \\geq k+1$. In this paper, we propose the concepts of kernel and closure of a graph and discuss the properties of closure. Utilizing these properties, we present the necessary and sufficient condition for a graph to be $k$-edge-maximal, which re","authors_text":"Hong-Jian Lai, Jian Lu, Zheng-Jiang Xia, Zhen-Mu Hong","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-30T13:05:05Z","title":"Characterization of the structure of $k$-edge-maximal graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00719","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":"sha256:9bec938d34476f860de91a3087cfcc1601f502d58ae39c26b6ba3dcd9aeb9a55","target":"record","created_at":"2026-06-02T01:04:03Z","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":"7fb993b74e09f1f74fa7f1ca74c529e1c0600b92aabcb13597f2b67e6177f096","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-30T13:05:05Z","title_canon_sha256":"2ed3761f90cc3437fc82e2aee4bedc8add5fe3cb1088eb04167a92b7ca7ed9b7"},"schema_version":"1.0","source":{"id":"2606.00719","kind":"arxiv","version":1}},"canonical_sha256":"8ce6b0660620f587a8a9856177913ef3f9d550fa8fd0936404ab7ee8d760b86e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ce6b0660620f587a8a9856177913ef3f9d550fa8fd0936404ab7ee8d760b86e","first_computed_at":"2026-06-02T01:04:03.696936Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T01:04:03.696936Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fX4bgde3mBzC69MGSclrcW7M5yvGp4U82f1Y4EvbuSqqkq9ew7MZSYCYMFO78JOXFyaLBAYNLu0JXQWwc1+nCw==","signature_status":"signed_v1","signed_at":"2026-06-02T01:04:03.697342Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.00719","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9bec938d34476f860de91a3087cfcc1601f502d58ae39c26b6ba3dcd9aeb9a55","sha256:0cb5aa7ed70e84316efa3a664660a7046934b7b0bea03e582de6e71ae9d1f87a"],"state_sha256":"512ec00f4ddc2bcef9e32c5f75630198a44c8418ea24076ecdedc1422e16fef4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8+zrdNMFSQfadeETGCmh8xXK8u/oRYOsWnl4fYDqmPKrHcZOYwvUaV+SESsymAdgV5Y+fi6tJQKwL6ESNT9+Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T19:58:21.227054Z","bundle_sha256":"d90847ef89e57ddfe950f7c2ffe200af90eef50b0454aa8c37cc348ddf95b91a"}}