{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6C7ZHFTPO2TPDSSOEDOEUCJJEA","short_pith_number":"pith:6C7ZHFTP","canonical_record":{"source":{"id":"1512.04200","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-14T07:05:04Z","cross_cats_sorted":[],"title_canon_sha256":"b49942cfb8ae8ed2e760153e154ceb70b26f052c8855798f4f06ccbf4bf16e40","abstract_canon_sha256":"8020649e286a3caa0d49bc202d9290f63d53b73342ba88cd399d2478eaf3252c"},"schema_version":"1.0"},"canonical_sha256":"f0bf93966f76a6f1ca4e20dc4a0929201672dbc62727e48ffa55d03f90dc333a","source":{"kind":"arxiv","id":"1512.04200","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04200","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04200v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04200","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"6C7ZHFTPO2TP","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6C7ZHFTPO2TPDSSO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6C7ZHFTP","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6C7ZHFTPO2TPDSSOEDOEUCJJEA","target":"record","payload":{"canonical_record":{"source":{"id":"1512.04200","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-14T07:05:04Z","cross_cats_sorted":[],"title_canon_sha256":"b49942cfb8ae8ed2e760153e154ceb70b26f052c8855798f4f06ccbf4bf16e40","abstract_canon_sha256":"8020649e286a3caa0d49bc202d9290f63d53b73342ba88cd399d2478eaf3252c"},"schema_version":"1.0"},"canonical_sha256":"f0bf93966f76a6f1ca4e20dc4a0929201672dbc62727e48ffa55d03f90dc333a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:22.862653Z","signature_b64":"pmay8hn4nt6aXINhtWhLgL6BHLQ2ODOvDbGqXKNpYDxY8gItV6yhVFb3+dBXXuyMCq4LjRk1F+qsfcHx0NiQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0bf93966f76a6f1ca4e20dc4a0929201672dbc62727e48ffa55d03f90dc333a","last_reissued_at":"2026-05-18T01:24:22.862051Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:22.862051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.04200","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":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GSf2GLShn/Dye2DYypdh6fLFCeK1Asbj7OX34oS1SzcLknkN6JAlmyfjYbbIRAQL1CBKlqKUuTpEss8cLPfdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T11:27:10.272536Z"},"content_sha256":"307a767da3d94040c02c74011bb0b3b78404e4afe07dca4e54ede5194d351aad","schema_version":"1.0","event_id":"sha256:307a767da3d94040c02c74011bb0b3b78404e4afe07dca4e54ede5194d351aad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6C7ZHFTPO2TPDSSOEDOEUCJJEA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parameterized Algorithms on Perfect Graphs for deletion to $(r,\\ell)$-graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Fahad Panolan, Saket Saurabh, Sudeshna Kolay, Venkatesh Raman","submitted_at":"2015-12-14T07:05:04Z","abstract_excerpt":"For fixed integers $r,\\ell \\geq 0$, a graph $G$ is called an {\\em $(r,\\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\\ell$ cliques. The class of $(r, \\ell)$ graphs generalizes $r$-colourable graphs (when $\\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \\ell)$-graph is \\NP-hard even when $r \\geq 3$ or $\\ell \\geq 3$ in general graphs.\n  When $r$ and $\\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\\sc Chromatic Number} problem is solvable in pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04200","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":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bOT7ec94CiyahG41KHq1N3zFjSU/uppNfPQ8f+PJjKi2Zbyt3d5JcQYSwdqb5YyOodCHvCJb+5Bx9WzJFIWTAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T11:27:10.273155Z"},"content_sha256":"677e43c5969eccd9e94273ed44ce8cd7946f0ee279a03c350cac870f8ad87ba0","schema_version":"1.0","event_id":"sha256:677e43c5969eccd9e94273ed44ce8cd7946f0ee279a03c350cac870f8ad87ba0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/bundle.json","state_url":"https://pith.science/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/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":"2026-05-22T11:27:10Z","links":{"resolver":"https://pith.science/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA","bundle":"https://pith.science/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/bundle.json","state":"https://pith.science/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6C7ZHFTPO2TPDSSOEDOEUCJJEA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6C7ZHFTPO2TPDSSOEDOEUCJJEA","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":"8020649e286a3caa0d49bc202d9290f63d53b73342ba88cd399d2478eaf3252c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-14T07:05:04Z","title_canon_sha256":"b49942cfb8ae8ed2e760153e154ceb70b26f052c8855798f4f06ccbf4bf16e40"},"schema_version":"1.0","source":{"id":"1512.04200","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04200","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04200v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04200","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"6C7ZHFTPO2TP","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6C7ZHFTPO2TPDSSO","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6C7ZHFTP","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:677e43c5969eccd9e94273ed44ce8cd7946f0ee279a03c350cac870f8ad87ba0","target":"graph","created_at":"2026-05-18T01:24:22Z","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 fixed integers $r,\\ell \\geq 0$, a graph $G$ is called an {\\em $(r,\\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\\ell$ cliques. The class of $(r, \\ell)$ graphs generalizes $r$-colourable graphs (when $\\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \\ell)$-graph is \\NP-hard even when $r \\geq 3$ or $\\ell \\geq 3$ in general graphs.\n  When $r$ and $\\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\\sc Chromatic Number} problem is solvable in pol","authors_text":"Fahad Panolan, Saket Saurabh, Sudeshna Kolay, Venkatesh Raman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-14T07:05:04Z","title":"Parameterized Algorithms on Perfect Graphs for deletion to $(r,\\ell)$-graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04200","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":"sha256:307a767da3d94040c02c74011bb0b3b78404e4afe07dca4e54ede5194d351aad","target":"record","created_at":"2026-05-18T01:24:22Z","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":"8020649e286a3caa0d49bc202d9290f63d53b73342ba88cd399d2478eaf3252c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-14T07:05:04Z","title_canon_sha256":"b49942cfb8ae8ed2e760153e154ceb70b26f052c8855798f4f06ccbf4bf16e40"},"schema_version":"1.0","source":{"id":"1512.04200","kind":"arxiv","version":1}},"canonical_sha256":"f0bf93966f76a6f1ca4e20dc4a0929201672dbc62727e48ffa55d03f90dc333a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0bf93966f76a6f1ca4e20dc4a0929201672dbc62727e48ffa55d03f90dc333a","first_computed_at":"2026-05-18T01:24:22.862051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:22.862051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pmay8hn4nt6aXINhtWhLgL6BHLQ2ODOvDbGqXKNpYDxY8gItV6yhVFb3+dBXXuyMCq4LjRk1F+qsfcHx0NiQBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:22.862653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.04200","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:307a767da3d94040c02c74011bb0b3b78404e4afe07dca4e54ede5194d351aad","sha256:677e43c5969eccd9e94273ed44ce8cd7946f0ee279a03c350cac870f8ad87ba0"],"state_sha256":"2fd167ef9bf2d8e57e93aa0e2b61a74f66e2a36356b69659999c9c92b4faf96b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"09Gx2l+CxI9AQ5TDXtPBbo56W9q4mYa84Oehncz5np2O/ya7PM8B1/RH39pDOw6UpypgSX58zno7rQmvSOwFCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T11:27:10.276002Z","bundle_sha256":"7f55f88c319230223d798071537715eff568ee8c856ea8bef2f07ada2cd1bae1"}}