{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ZWPPHOEUFJLWT5OYS623UXGXGW","short_pith_number":"pith:ZWPPHOEU","canonical_record":{"source":{"id":"1512.01113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-12-02T10:53:08Z","cross_cats_sorted":[],"title_canon_sha256":"ec1c976a2cf7b74733b519856759972955927a752ca4d6eca5acf10176ad6637","abstract_canon_sha256":"c33266669d863cf789b3ee9d7cf706b6160a80657262dd36be8ee758a7ecc600"},"schema_version":"1.0"},"canonical_sha256":"cd9ef3b8942a5769f5d897b5ba5cd7359d6fd825efe28ae57ad173b6abb8bd20","source":{"kind":"arxiv","id":"1512.01113","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01113","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01113v1","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01113","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZWPPHOEUFJLW","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZWPPHOEUFJLWT5OY","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZWPPHOEU","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ZWPPHOEUFJLWT5OYS623UXGXGW","target":"record","payload":{"canonical_record":{"source":{"id":"1512.01113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-12-02T10:53:08Z","cross_cats_sorted":[],"title_canon_sha256":"ec1c976a2cf7b74733b519856759972955927a752ca4d6eca5acf10176ad6637","abstract_canon_sha256":"c33266669d863cf789b3ee9d7cf706b6160a80657262dd36be8ee758a7ecc600"},"schema_version":"1.0"},"canonical_sha256":"cd9ef3b8942a5769f5d897b5ba5cd7359d6fd825efe28ae57ad173b6abb8bd20","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:18.925487Z","signature_b64":"xByZa6e4SXPFpJrUFTSqvNvVcZCh8wPTGMm5uBd907jFr6Cqj8zQlLT4IV4ATT0eMvJG3teLYxbEk7to8dIqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd9ef3b8942a5769f5d897b5ba5cd7359d6fd825efe28ae57ad173b6abb8bd20","last_reissued_at":"2026-05-18T01:25:18.924815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:18.924815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.01113","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":"2026-05-18T01:25:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ywP0E5dCch5sEvHbPlxCJOI2GgbhkC8uZrn3Cbly8V21vfFFHU0efeahXayGgFXtubQ9nCBUTFEa5shCern9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T20:18:30.581458Z"},"content_sha256":"f5f76bcb90ddc32c6c213ed5af2ee606a7daa802cdcb8f72fc57f6c684d6833e","schema_version":"1.0","event_id":"sha256:f5f76bcb90ddc32c6c213ed5af2ee606a7daa802cdcb8f72fc57f6c684d6833e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ZWPPHOEUFJLWT5OYS623UXGXGW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Note on Sparing Number Algorithm of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"K. A. Germina, N. K. Sudev","submitted_at":"2015-12-02T10:53:08Z","abstract_excerpt":"Let $X$ denote a set of all non-negative integers and $\\sP(X)$ be its power set. A weak integer additive set-labeling (WIASL) of a graph $G$ is an injective set-valued function $f:V(G)\\to \\sP(X)-\\{\\emptyset\\}$ where induced function $f^+:E(G) \\to \\sP(X)-\\{\\emptyset\\}$ is defined by $f^+ (uv) = f(u)+ f(v)$ such that either $|f^+ (uv)|=|f(u)|$ or $|f^+ (uv)|=|f(v)|$ , where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. The sparing number of a WIASL-graph $G$ is the minimum required number of edges in $G$ having singleton set-labels. In this paper, we discuss an algorithm for finding the sparin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01113","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":""},"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":"2026-05-18T01:25:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AFoq9vQ2zuEQh8MJeGcPo9zGLcxECe/eOMuZFVWdrFUGdOiM0JTeLwnGFOzET8LAIh+6Q8e3YklHJDLtZhdYBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T20:18:30.582134Z"},"content_sha256":"0f04e9ad110e10658eee3db7797102bd6492a0c47b818dd9a2235ce5f81c1381","schema_version":"1.0","event_id":"sha256:0f04e9ad110e10658eee3db7797102bd6492a0c47b818dd9a2235ce5f81c1381"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/bundle.json","state_url":"https://pith.science/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/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":"2026-06-11T20:18:30Z","links":{"resolver":"https://pith.science/pith/ZWPPHOEUFJLWT5OYS623UXGXGW","bundle":"https://pith.science/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/bundle.json","state":"https://pith.science/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZWPPHOEUFJLWT5OYS623UXGXGW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZWPPHOEUFJLWT5OYS623UXGXGW","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":"c33266669d863cf789b3ee9d7cf706b6160a80657262dd36be8ee758a7ecc600","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-12-02T10:53:08Z","title_canon_sha256":"ec1c976a2cf7b74733b519856759972955927a752ca4d6eca5acf10176ad6637"},"schema_version":"1.0","source":{"id":"1512.01113","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01113","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01113v1","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01113","created_at":"2026-05-18T01:25:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZWPPHOEUFJLW","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZWPPHOEUFJLWT5OY","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZWPPHOEU","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:0f04e9ad110e10658eee3db7797102bd6492a0c47b818dd9a2235ce5f81c1381","target":"graph","created_at":"2026-05-18T01:25:18Z","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":"Let $X$ denote a set of all non-negative integers and $\\sP(X)$ be its power set. A weak integer additive set-labeling (WIASL) of a graph $G$ is an injective set-valued function $f:V(G)\\to \\sP(X)-\\{\\emptyset\\}$ where induced function $f^+:E(G) \\to \\sP(X)-\\{\\emptyset\\}$ is defined by $f^+ (uv) = f(u)+ f(v)$ such that either $|f^+ (uv)|=|f(u)|$ or $|f^+ (uv)|=|f(v)|$ , where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. The sparing number of a WIASL-graph $G$ is the minimum required number of edges in $G$ having singleton set-labels. In this paper, we discuss an algorithm for finding the sparin","authors_text":"K. A. Germina, N. K. Sudev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-12-02T10:53:08Z","title":"A Note on Sparing Number Algorithm of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01113","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:f5f76bcb90ddc32c6c213ed5af2ee606a7daa802cdcb8f72fc57f6c684d6833e","target":"record","created_at":"2026-05-18T01:25:18Z","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":"c33266669d863cf789b3ee9d7cf706b6160a80657262dd36be8ee758a7ecc600","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2015-12-02T10:53:08Z","title_canon_sha256":"ec1c976a2cf7b74733b519856759972955927a752ca4d6eca5acf10176ad6637"},"schema_version":"1.0","source":{"id":"1512.01113","kind":"arxiv","version":1}},"canonical_sha256":"cd9ef3b8942a5769f5d897b5ba5cd7359d6fd825efe28ae57ad173b6abb8bd20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd9ef3b8942a5769f5d897b5ba5cd7359d6fd825efe28ae57ad173b6abb8bd20","first_computed_at":"2026-05-18T01:25:18.924815Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:18.924815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xByZa6e4SXPFpJrUFTSqvNvVcZCh8wPTGMm5uBd907jFr6Cqj8zQlLT4IV4ATT0eMvJG3teLYxbEk7to8dIqAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:18.925487Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.01113","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5f76bcb90ddc32c6c213ed5af2ee606a7daa802cdcb8f72fc57f6c684d6833e","sha256:0f04e9ad110e10658eee3db7797102bd6492a0c47b818dd9a2235ce5f81c1381"],"state_sha256":"62288f27064646139d59bf4e17e8ca628a567c20612d798444571f6202b12fef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CnulNSV50QGpm4DEVJPYu6Ij6+cWwvHusZwbfgI2ugfAKN73gVI8WA2y3OG0NdY7kDV4fZEJXD4gd2Z4FkssCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T20:18:30.587094Z","bundle_sha256":"6e8493c453fd93d23a38bf069dedeef18754f6f6bc88053c76e030ce998f7b9e"}}