{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:H2MGLROIPHQD2IXIJISSVCEKR2","short_pith_number":"pith:H2MGLROI","canonical_record":{"source":{"id":"1603.09532","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-31T11:34:48Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"50f52d6625e36a123bc9f7a8e880fed53dd4b23c6b8ac76f0ce7f85ad400833c","abstract_canon_sha256":"461cc721db828040d0563805af9fcf731ea5c14dc027fa058e8b7dc57dae14cd"},"schema_version":"1.0"},"canonical_sha256":"3e9865c5c879e03d22e84a252a888a8e8d5885120e8eb63381b783d69cc8fdbd","source":{"kind":"arxiv","id":"1603.09532","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.09532","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"arxiv_version","alias_value":"1603.09532v2","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.09532","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"pith_short_12","alias_value":"H2MGLROIPHQD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"H2MGLROIPHQD2IXI","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"H2MGLROI","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:H2MGLROIPHQD2IXIJISSVCEKR2","target":"record","payload":{"canonical_record":{"source":{"id":"1603.09532","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-31T11:34:48Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"50f52d6625e36a123bc9f7a8e880fed53dd4b23c6b8ac76f0ce7f85ad400833c","abstract_canon_sha256":"461cc721db828040d0563805af9fcf731ea5c14dc027fa058e8b7dc57dae14cd"},"schema_version":"1.0"},"canonical_sha256":"3e9865c5c879e03d22e84a252a888a8e8d5885120e8eb63381b783d69cc8fdbd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:39.467711Z","signature_b64":"mVqqbZmMORbhjqx3TFLAGZe/YahhPG4OZQSp5blvJmzublJNDJSdYeFuB5gPUDjg1//ostKkcNaU+rJ6lDgjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e9865c5c879e03d22e84a252a888a8e8d5885120e8eb63381b783d69cc8fdbd","last_reissued_at":"2026-05-18T01:00:39.467271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:39.467271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.09532","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:00:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ogjHqGLgAFpIeBpTmZLVn5T0e8kOKM2MRqO1P7LKaUZtwehM2FHCp85l1fdnVdCe9P8p35Pc1Q08hYx4a3eqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:24:21.688136Z"},"content_sha256":"7470b318975069c53fbeca821e13380022fda4a2425973f40a104250dcb5d8b8","schema_version":"1.0","event_id":"sha256:7470b318975069c53fbeca821e13380022fda4a2425973f40a104250dcb5d8b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:H2MGLROIPHQD2IXIJISSVCEKR2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Characterising Bounded Expansion by Neighbourhood Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Felix Reidl, Fernando S\\'anchez Villaamil, Konstantinos Stavropoulos","submitted_at":"2016-03-31T11:34:48Z","abstract_excerpt":"We show that a graph class $\\cal G$ has bounded expansion if and only if it has bounded $r$-neighbourhood complexity, i.e. for any vertex set $X$ of any subgraph $H$ of $G\\in\\cal G$, the number of subsets of $X$ which are exact $r$-neighbourhoods of vertices of $H$ on $X$ is linear to the size of $X$. This is established by bounding the $r$-neighbourhood complexity of a graph in terms of both its $r$-centred colouring number and its weak $r$-colouring number, which provide known characterisations to the property of bounded expansion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09532","kind":"arxiv","version":2},"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":"2026-05-18T01:00:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xcy3d48Uz/mRZMkpvDRBLbuYcU3PpSgozJMXpsmjmmgA7K+KeBZO4YxQ5CzIHbIGAk40ESiJQxASEKLv29HABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:24:21.688512Z"},"content_sha256":"007a0ccb8504f9993d12a244e5cd7b9491d1818ba7ac86d311fe5d8b16b26ed8","schema_version":"1.0","event_id":"sha256:007a0ccb8504f9993d12a244e5cd7b9491d1818ba7ac86d311fe5d8b16b26ed8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H2MGLROIPHQD2IXIJISSVCEKR2/bundle.json","state_url":"https://pith.science/pith/H2MGLROIPHQD2IXIJISSVCEKR2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H2MGLROIPHQD2IXIJISSVCEKR2/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":"2026-05-30T09:24:21Z","links":{"resolver":"https://pith.science/pith/H2MGLROIPHQD2IXIJISSVCEKR2","bundle":"https://pith.science/pith/H2MGLROIPHQD2IXIJISSVCEKR2/bundle.json","state":"https://pith.science/pith/H2MGLROIPHQD2IXIJISSVCEKR2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H2MGLROIPHQD2IXIJISSVCEKR2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:H2MGLROIPHQD2IXIJISSVCEKR2","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":"461cc721db828040d0563805af9fcf731ea5c14dc027fa058e8b7dc57dae14cd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-31T11:34:48Z","title_canon_sha256":"50f52d6625e36a123bc9f7a8e880fed53dd4b23c6b8ac76f0ce7f85ad400833c"},"schema_version":"1.0","source":{"id":"1603.09532","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.09532","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"arxiv_version","alias_value":"1603.09532v2","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.09532","created_at":"2026-05-18T01:00:39Z"},{"alias_kind":"pith_short_12","alias_value":"H2MGLROIPHQD","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"H2MGLROIPHQD2IXI","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"H2MGLROI","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:007a0ccb8504f9993d12a244e5cd7b9491d1818ba7ac86d311fe5d8b16b26ed8","target":"graph","created_at":"2026-05-18T01:00:39Z","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":"We show that a graph class $\\cal G$ has bounded expansion if and only if it has bounded $r$-neighbourhood complexity, i.e. for any vertex set $X$ of any subgraph $H$ of $G\\in\\cal G$, the number of subsets of $X$ which are exact $r$-neighbourhoods of vertices of $H$ on $X$ is linear to the size of $X$. This is established by bounding the $r$-neighbourhood complexity of a graph in terms of both its $r$-centred colouring number and its weak $r$-colouring number, which provide known characterisations to the property of bounded expansion.","authors_text":"Felix Reidl, Fernando S\\'anchez Villaamil, Konstantinos Stavropoulos","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-31T11:34:48Z","title":"Characterising Bounded Expansion by Neighbourhood Complexity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09532","kind":"arxiv","version":2},"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":"sha256:7470b318975069c53fbeca821e13380022fda4a2425973f40a104250dcb5d8b8","target":"record","created_at":"2026-05-18T01:00:39Z","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":"461cc721db828040d0563805af9fcf731ea5c14dc027fa058e8b7dc57dae14cd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-31T11:34:48Z","title_canon_sha256":"50f52d6625e36a123bc9f7a8e880fed53dd4b23c6b8ac76f0ce7f85ad400833c"},"schema_version":"1.0","source":{"id":"1603.09532","kind":"arxiv","version":2}},"canonical_sha256":"3e9865c5c879e03d22e84a252a888a8e8d5885120e8eb63381b783d69cc8fdbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e9865c5c879e03d22e84a252a888a8e8d5885120e8eb63381b783d69cc8fdbd","first_computed_at":"2026-05-18T01:00:39.467271Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:39.467271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mVqqbZmMORbhjqx3TFLAGZe/YahhPG4OZQSp5blvJmzublJNDJSdYeFuB5gPUDjg1//ostKkcNaU+rJ6lDgjAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:39.467711Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.09532","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7470b318975069c53fbeca821e13380022fda4a2425973f40a104250dcb5d8b8","sha256:007a0ccb8504f9993d12a244e5cd7b9491d1818ba7ac86d311fe5d8b16b26ed8"],"state_sha256":"94075fd0158393dfa3e9c970102f2ce5bc9eab7f6ee0356b848be82fdc5c24d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JtvfVZbfp2meMrT+5XrjOPBT7pwJLw7TPkh18m6RlJZgBL6T5t6AGiiGHE4Vq1PNw2UAXylDpq/yKL5mtrfpDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T09:24:21.690523Z","bundle_sha256":"6fe19de9c15e160da48f890514e8efd37825cee75171100222509f7d0e0b847e"}}