{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:24HSHBMYRXY5MIGHZ6CJP4Y3MF","short_pith_number":"pith:24HSHBMY","canonical_record":{"source":{"id":"1512.01781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-06T12:09:52Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fa6fb386b061fea8fe1a54cdc4ab846d3937fd687044a2b767932bd27614c25f","abstract_canon_sha256":"6bd49673bac4e65e6f281e77da0273a9badb33212b8fb0d07be5139735c23851"},"schema_version":"1.0"},"canonical_sha256":"d70f2385988df1d620c7cf8497f31b614f1f31774fef4d9ee10daa5af6c6fe93","source":{"kind":"arxiv","id":"1512.01781","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01781","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01781v1","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01781","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"24HSHBMYRXY5","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"24HSHBMYRXY5MIGH","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"24HSHBMY","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:24HSHBMYRXY5MIGHZ6CJP4Y3MF","target":"record","payload":{"canonical_record":{"source":{"id":"1512.01781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-06T12:09:52Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"fa6fb386b061fea8fe1a54cdc4ab846d3937fd687044a2b767932bd27614c25f","abstract_canon_sha256":"6bd49673bac4e65e6f281e77da0273a9badb33212b8fb0d07be5139735c23851"},"schema_version":"1.0"},"canonical_sha256":"d70f2385988df1d620c7cf8497f31b614f1f31774fef4d9ee10daa5af6c6fe93","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:10.380136Z","signature_b64":"c3XZ6ZP866VuQZ6R1JTh61ordkWfnJjoBjGf2QnlknM7j+bs2K5MjlPQ1GxK6XwWc1MLdbB1UgIJCyfzsWjUDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d70f2385988df1d620c7cf8497f31b614f1f31774fef4d9ee10daa5af6c6fe93","last_reissued_at":"2026-05-18T01:25:10.379464Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:10.379464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.01781","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":"2026-05-18T01:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rT20XKVRmVwmFLKNlDAPFHKII7t2eh/EKlYVAWVUiGZ1HfGGqujpsGoHCGO2/GZ5k9tsWnfByxv8IQi0tAArCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:04:35.024186Z"},"content_sha256":"5c342f5d1a9d65d7476879c925315ab1b6e0e633e8a4a35860c4495dabcefc2b","schema_version":"1.0","event_id":"sha256:5c342f5d1a9d65d7476879c925315ab1b6e0e633e8a4a35860c4495dabcefc2b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:24HSHBMYRXY5MIGHZ6CJP4Y3MF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"k-Trails: Recognition, Complexity, and Approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Mohit Singh, Rico Zenklusen","submitted_at":"2015-12-06T12:09:52Z","abstract_excerpt":"The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails.\n  In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve sev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01781","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":"2026-05-18T01:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pCk4lIwX3pOdIbvhGQRfRVBeXHxDPkr/F7l0CmqntJbApZde7G9sCPceMtCzbClemDShhUxOT3tLMdBobKubCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:04:35.024906Z"},"content_sha256":"9085bb6167746b8ce392243f4bb6630a3584b331ad5efa6234948d5c1ccaed46","schema_version":"1.0","event_id":"sha256:9085bb6167746b8ce392243f4bb6630a3584b331ad5efa6234948d5c1ccaed46"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/bundle.json","state_url":"https://pith.science/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/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":"2026-05-31T11:04:35Z","links":{"resolver":"https://pith.science/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF","bundle":"https://pith.science/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/bundle.json","state":"https://pith.science/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24HSHBMYRXY5MIGHZ6CJP4Y3MF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:24HSHBMYRXY5MIGHZ6CJP4Y3MF","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":"6bd49673bac4e65e6f281e77da0273a9badb33212b8fb0d07be5139735c23851","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-06T12:09:52Z","title_canon_sha256":"fa6fb386b061fea8fe1a54cdc4ab846d3937fd687044a2b767932bd27614c25f"},"schema_version":"1.0","source":{"id":"1512.01781","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01781","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01781v1","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01781","created_at":"2026-05-18T01:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"24HSHBMYRXY5","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"24HSHBMYRXY5MIGH","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"24HSHBMY","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:9085bb6167746b8ce392243f4bb6630a3584b331ad5efa6234948d5c1ccaed46","target":"graph","created_at":"2026-05-18T01:25:10Z","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":"The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails.\n  In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve sev","authors_text":"Mohit Singh, Rico Zenklusen","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-06T12:09:52Z","title":"k-Trails: Recognition, Complexity, and Approximations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01781","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":"sha256:5c342f5d1a9d65d7476879c925315ab1b6e0e633e8a4a35860c4495dabcefc2b","target":"record","created_at":"2026-05-18T01:25:10Z","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":"6bd49673bac4e65e6f281e77da0273a9badb33212b8fb0d07be5139735c23851","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-12-06T12:09:52Z","title_canon_sha256":"fa6fb386b061fea8fe1a54cdc4ab846d3937fd687044a2b767932bd27614c25f"},"schema_version":"1.0","source":{"id":"1512.01781","kind":"arxiv","version":1}},"canonical_sha256":"d70f2385988df1d620c7cf8497f31b614f1f31774fef4d9ee10daa5af6c6fe93","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d70f2385988df1d620c7cf8497f31b614f1f31774fef4d9ee10daa5af6c6fe93","first_computed_at":"2026-05-18T01:25:10.379464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:10.379464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c3XZ6ZP866VuQZ6R1JTh61ordkWfnJjoBjGf2QnlknM7j+bs2K5MjlPQ1GxK6XwWc1MLdbB1UgIJCyfzsWjUDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:10.380136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.01781","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c342f5d1a9d65d7476879c925315ab1b6e0e633e8a4a35860c4495dabcefc2b","sha256:9085bb6167746b8ce392243f4bb6630a3584b331ad5efa6234948d5c1ccaed46"],"state_sha256":"f1e0c2cb9cc1f6456e9730a1ae1c3b6e038fec2e97c8b51023488f57f869445c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B3nG+rP7bNLTQ/xqCbS5bAOMTNtnKkJejG3vAA+8CjK6IVyvY2ujTrbLM5qdAKT18lLMjWaN+2FqNP9nR5CzAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T11:04:35.028496Z","bundle_sha256":"ffe7845e94a88316b2ff1343305286b6dba45952f4ae13c3d51b24e2cb3a9165"}}