{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:IY6OGDLYQUQEGH577BSDXNCKF7","short_pith_number":"pith:IY6OGDLY","canonical_record":{"source":{"id":"1312.3182","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-12-11T14:20:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b885d17741e86fc39247f7d7dcba9cbe35f8b690694cfce9cad4b4d266a35b0a","abstract_canon_sha256":"18c11bbca6e921f7e62cf705db0435ed366d31b0520e3851abad7c20b5f024b1"},"schema_version":"1.0"},"canonical_sha256":"463ce30d788520431fbff8643bb44a2fff6c6150434566ae12a354e4ce6e4036","source":{"kind":"arxiv","id":"1312.3182","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.3182","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1312.3182v1","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3182","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"IY6OGDLYQUQE","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"IY6OGDLYQUQEGH57","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"IY6OGDLY","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:IY6OGDLYQUQEGH577BSDXNCKF7","target":"record","payload":{"canonical_record":{"source":{"id":"1312.3182","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-12-11T14:20:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b885d17741e86fc39247f7d7dcba9cbe35f8b690694cfce9cad4b4d266a35b0a","abstract_canon_sha256":"18c11bbca6e921f7e62cf705db0435ed366d31b0520e3851abad7c20b5f024b1"},"schema_version":"1.0"},"canonical_sha256":"463ce30d788520431fbff8643bb44a2fff6c6150434566ae12a354e4ce6e4036","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:00.277178Z","signature_b64":"g8qIxzH5Xrw8s9+dYtJSedH6DnxaRf13k7ojX6YpJCarVUbz0t8ODbJc903n3LY7Rgp/R33B3wBZWhs3JOz/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"463ce30d788520431fbff8643bb44a2fff6c6150434566ae12a354e4ce6e4036","last_reissued_at":"2026-05-18T03:05:00.276672Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:00.276672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.3182","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":"2026-05-18T03:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8CRshoZdWsK2RUEb/AnMxFcUsAQO/14OHEwZSGX8Am8VpBtDmtIfXDH7car3ADJV07V4s09W7QbV9EJIbJVlAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:24:13.115352Z"},"content_sha256":"115b5c009ee29bf98cc1fb106ca95f9912eb4607a6d796e6d2b53dff5a3a1c32","schema_version":"1.0","event_id":"sha256:115b5c009ee29bf98cc1fb106ca95f9912eb4607a6d796e6d2b53dff5a3a1c32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:IY6OGDLYQUQEGH577BSDXNCKF7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On The Center Sets and Center Numbers of Some Graph Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"A.Sreekumar, Kannan Balakrishnan, Manoj Changat, Prasanth G. Narasimha-Shenoi, Ram Kumar. R","submitted_at":"2013-12-11T14:20:55Z","abstract_excerpt":"For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\\displaystyle\\max_{x \\in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3182","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":"2026-05-18T03:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tQN/PixgrDXbsVVfvWKuJBkaCFE6SMC1WDTBrixshlo57G4kYWtyZe72novvnitYMN4xr8w1BG/Tx2nGcAcKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:24:13.115721Z"},"content_sha256":"9917b142ff748d970c35446e0d58cd12fd29974227cd58a0fa1cd1f668aa53dc","schema_version":"1.0","event_id":"sha256:9917b142ff748d970c35446e0d58cd12fd29974227cd58a0fa1cd1f668aa53dc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IY6OGDLYQUQEGH577BSDXNCKF7/bundle.json","state_url":"https://pith.science/pith/IY6OGDLYQUQEGH577BSDXNCKF7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IY6OGDLYQUQEGH577BSDXNCKF7/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":"2026-05-28T16:24:13Z","links":{"resolver":"https://pith.science/pith/IY6OGDLYQUQEGH577BSDXNCKF7","bundle":"https://pith.science/pith/IY6OGDLYQUQEGH577BSDXNCKF7/bundle.json","state":"https://pith.science/pith/IY6OGDLYQUQEGH577BSDXNCKF7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IY6OGDLYQUQEGH577BSDXNCKF7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IY6OGDLYQUQEGH577BSDXNCKF7","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":"18c11bbca6e921f7e62cf705db0435ed366d31b0520e3851abad7c20b5f024b1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-12-11T14:20:55Z","title_canon_sha256":"b885d17741e86fc39247f7d7dcba9cbe35f8b690694cfce9cad4b4d266a35b0a"},"schema_version":"1.0","source":{"id":"1312.3182","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.3182","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1312.3182v1","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3182","created_at":"2026-05-18T03:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"IY6OGDLYQUQE","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"IY6OGDLYQUQEGH57","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"IY6OGDLY","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:9917b142ff748d970c35446e0d58cd12fd29974227cd58a0fa1cd1f668aa53dc","target":"graph","created_at":"2026-05-18T03:05:00Z","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":"For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\\displaystyle\\max_{x \\in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center ","authors_text":"A.Sreekumar, Kannan Balakrishnan, Manoj Changat, Prasanth G. Narasimha-Shenoi, Ram Kumar. R","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-12-11T14:20:55Z","title":"On The Center Sets and Center Numbers of Some Graph Classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3182","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":"sha256:115b5c009ee29bf98cc1fb106ca95f9912eb4607a6d796e6d2b53dff5a3a1c32","target":"record","created_at":"2026-05-18T03:05:00Z","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":"18c11bbca6e921f7e62cf705db0435ed366d31b0520e3851abad7c20b5f024b1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2013-12-11T14:20:55Z","title_canon_sha256":"b885d17741e86fc39247f7d7dcba9cbe35f8b690694cfce9cad4b4d266a35b0a"},"schema_version":"1.0","source":{"id":"1312.3182","kind":"arxiv","version":1}},"canonical_sha256":"463ce30d788520431fbff8643bb44a2fff6c6150434566ae12a354e4ce6e4036","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"463ce30d788520431fbff8643bb44a2fff6c6150434566ae12a354e4ce6e4036","first_computed_at":"2026-05-18T03:05:00.276672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:00.276672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g8qIxzH5Xrw8s9+dYtJSedH6DnxaRf13k7ojX6YpJCarVUbz0t8ODbJc903n3LY7Rgp/R33B3wBZWhs3JOz/AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:00.277178Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.3182","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:115b5c009ee29bf98cc1fb106ca95f9912eb4607a6d796e6d2b53dff5a3a1c32","sha256:9917b142ff748d970c35446e0d58cd12fd29974227cd58a0fa1cd1f668aa53dc"],"state_sha256":"857a9d7ddca04bd190aa15bf86664e31f05bbb0aa70a338f68c61575597b635c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pIeZ0ZQl48VHxV4L9OZQDmaQ19yCYrQyMEOfy6dArxmneKTaSOHRjQiySzo2aX5U732+/e6VJO7liKgsEJfGAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:24:13.117712Z","bundle_sha256":"647b4f57dce35891642737f769c39ef85e96366314a0d3d473b8270b2b8a4e0b"}}