{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:S3GNBWKXAVS2LT4OS3XYT2UZ7I","short_pith_number":"pith:S3GNBWKX","canonical_record":{"source":{"id":"1311.1108","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-05T16:13:28Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"b68aaefd535f3ea328b6adc0556cdd93ee57df331fff5878db03b6992fb264e1","abstract_canon_sha256":"e5ac7e6c50bb42d34ff9b3c76ef0e41c7d558de987395eb9331cca5ee83f3010"},"schema_version":"1.0"},"canonical_sha256":"96ccd0d9570565a5cf8e96ef89ea99fa3d11a0aa2316f25348488963eaa2f375","source":{"kind":"arxiv","id":"1311.1108","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1108","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1108v3","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1108","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"pith_short_12","alias_value":"S3GNBWKXAVS2","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"S3GNBWKXAVS2LT4O","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"S3GNBWKX","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:S3GNBWKXAVS2LT4OS3XYT2UZ7I","target":"record","payload":{"canonical_record":{"source":{"id":"1311.1108","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-05T16:13:28Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"b68aaefd535f3ea328b6adc0556cdd93ee57df331fff5878db03b6992fb264e1","abstract_canon_sha256":"e5ac7e6c50bb42d34ff9b3c76ef0e41c7d558de987395eb9331cca5ee83f3010"},"schema_version":"1.0"},"canonical_sha256":"96ccd0d9570565a5cf8e96ef89ea99fa3d11a0aa2316f25348488963eaa2f375","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:06.998772Z","signature_b64":"dNZE6+d6hYsSRKkqtYbJNeMAoIPxtODpl5E3ZU9q/suHCvuiIBZT5DBgq0y9qAHb78v54lBrER+Dx3tuaRniDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96ccd0d9570565a5cf8e96ef89ea99fa3d11a0aa2316f25348488963eaa2f375","last_reissued_at":"2026-05-18T01:33:06.998346Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:06.998346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.1108","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:33:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5yBmIJnss2IIscnDGWXRiBvREIyMA5Ftm7WaKrCoFPxs1CzO3L3z6it8HFOgyAzVsdkCOXXRxmXb/shjG2jJBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:34:23.765748Z"},"content_sha256":"805edddceed7fda04604d844ff3f9b3bd7ae5636c31aadb01e1d5e3704d0c23d","schema_version":"1.0","event_id":"sha256:805edddceed7fda04604d844ff3f9b3bd7ae5636c31aadb01e1d5e3704d0c23d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:S3GNBWKXAVS2LT4OS3XYT2UZ7I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An edge variant of the Erd\\H{o}s-P\\'osa property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Dimitrios M. Thilikos, Ignasi Sau, Jean-Florent Raymond","submitted_at":"2013-11-05T16:13:28Z","abstract_excerpt":"For every $r\\in \\mathbb{N}$, we denote by $\\theta_{r}$ the multigraph with two vertices and $r$ parallel edges. Given a graph $G$, we say that a subgraph $H$ of $G$ is a model of $\\theta_{r}$ in $G$ if $H$ contains $\\theta_{r}$ as a contraction. We prove that the following edge variant of the Erd{\\H o}s-P{\\'o}sa property holds for every $r\\geq 2$: if $G$ is a graph and $k$ is a positive integer, then either $G$ contains a packing of $k$ mutually edge-disjoint models of $\\theta_{r}$, or it contains a set $S$ of $f_r(k)$ edges such that $G\\setminus S$ has no $\\theta_{r}$-model, for both $f_r(k) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1108","kind":"arxiv","version":3},"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":"2026-05-18T01:33:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L4kZpxnRGZXUgOWsavNE5h9dOZpxOAYu9+2h3WngE9W9ptk52lHux0NVAvx4L2dOBlP67OBHO3n3QZb+NayKCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:34:23.766196Z"},"content_sha256":"e867e74147760d0651fc99bf8c3259465aedec584046ce1bafe137a1d305eb14","schema_version":"1.0","event_id":"sha256:e867e74147760d0651fc99bf8c3259465aedec584046ce1bafe137a1d305eb14"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/bundle.json","state_url":"https://pith.science/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/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":"2026-05-25T16:34:23Z","links":{"resolver":"https://pith.science/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I","bundle":"https://pith.science/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/bundle.json","state":"https://pith.science/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S3GNBWKXAVS2LT4OS3XYT2UZ7I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:S3GNBWKXAVS2LT4OS3XYT2UZ7I","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":"e5ac7e6c50bb42d34ff9b3c76ef0e41c7d558de987395eb9331cca5ee83f3010","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-05T16:13:28Z","title_canon_sha256":"b68aaefd535f3ea328b6adc0556cdd93ee57df331fff5878db03b6992fb264e1"},"schema_version":"1.0","source":{"id":"1311.1108","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1108","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1108v3","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1108","created_at":"2026-05-18T01:33:06Z"},{"alias_kind":"pith_short_12","alias_value":"S3GNBWKXAVS2","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"S3GNBWKXAVS2LT4O","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"S3GNBWKX","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:e867e74147760d0651fc99bf8c3259465aedec584046ce1bafe137a1d305eb14","target":"graph","created_at":"2026-05-18T01:33:06Z","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 every $r\\in \\mathbb{N}$, we denote by $\\theta_{r}$ the multigraph with two vertices and $r$ parallel edges. Given a graph $G$, we say that a subgraph $H$ of $G$ is a model of $\\theta_{r}$ in $G$ if $H$ contains $\\theta_{r}$ as a contraction. We prove that the following edge variant of the Erd{\\H o}s-P{\\'o}sa property holds for every $r\\geq 2$: if $G$ is a graph and $k$ is a positive integer, then either $G$ contains a packing of $k$ mutually edge-disjoint models of $\\theta_{r}$, or it contains a set $S$ of $f_r(k)$ edges such that $G\\setminus S$ has no $\\theta_{r}$-model, for both $f_r(k) ","authors_text":"Dimitrios M. Thilikos, Ignasi Sau, Jean-Florent Raymond","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-05T16:13:28Z","title":"An edge variant of the Erd\\H{o}s-P\\'osa property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1108","kind":"arxiv","version":3},"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":"sha256:805edddceed7fda04604d844ff3f9b3bd7ae5636c31aadb01e1d5e3704d0c23d","target":"record","created_at":"2026-05-18T01:33:06Z","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":"e5ac7e6c50bb42d34ff9b3c76ef0e41c7d558de987395eb9331cca5ee83f3010","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-05T16:13:28Z","title_canon_sha256":"b68aaefd535f3ea328b6adc0556cdd93ee57df331fff5878db03b6992fb264e1"},"schema_version":"1.0","source":{"id":"1311.1108","kind":"arxiv","version":3}},"canonical_sha256":"96ccd0d9570565a5cf8e96ef89ea99fa3d11a0aa2316f25348488963eaa2f375","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96ccd0d9570565a5cf8e96ef89ea99fa3d11a0aa2316f25348488963eaa2f375","first_computed_at":"2026-05-18T01:33:06.998346Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:06.998346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dNZE6+d6hYsSRKkqtYbJNeMAoIPxtODpl5E3ZU9q/suHCvuiIBZT5DBgq0y9qAHb78v54lBrER+Dx3tuaRniDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:06.998772Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.1108","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:805edddceed7fda04604d844ff3f9b3bd7ae5636c31aadb01e1d5e3704d0c23d","sha256:e867e74147760d0651fc99bf8c3259465aedec584046ce1bafe137a1d305eb14"],"state_sha256":"2071c1661a7408b4b8df9e21e4e8d22e926b65959a018824a20b63a34052cfa3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qh+PVatkdV7j7gXqxhRxSIZVFuipxzqjnRimuRr5DC3+E1YiSf7ovImEq1DmBESPdhwUjiGBnKBdWktLCy4VAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:34:23.769980Z","bundle_sha256":"8ac99a4d968a852858fcdf7f3a9319a11d98ac303335add0207835ce70e06c00"}}