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Let $S_{t,t,t}$ be the graph obtained from $K_{1,3}$ by subdividing each edge $t-1$ times, and let $W_{t\\times t}$ be the $t$-by-$t$ hexagonal grid. Let $\\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \\times t}$. We prove that for every positive integer $t$ there exists $c(t)$ such that every $\\mathcal{L}_t \\cup \\{S_{t,t,t}, K_{t,t}\\}$-free $n$-vertex graph admits a tree decom"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2501.14658","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-24T17:27:37Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"bfbc597e73bc9d3abf67374045489b6f4c6923daa23bb615af152c95873b46b5","abstract_canon_sha256":"1c226e916aec0b0b578fc1f02f0225d16112ef02fada9852d394e4e16e9f7094"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:10:48.239107Z","signature_b64":"YHcpwBPEZQHOwI6Mz1ilaQlqAQh89NBu9SWFRhdxa5r6sxIkxJWwcCUrUpguWggRpIhPtHBvZENNBY12VWI4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3131404440562e77d6471928b357f238e35569453eee4b0a12213d588c41581a","last_reissued_at":"2026-07-05T10:10:48.238670Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:10:48.238670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tree independence number V. Walls and claws","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Daniel Lokshtanov, Julien Codsi, Maria Chudnovsky, Martin Milani\\v{c}, Varun Sivashankar","submitted_at":"2025-01-24T17:27:37Z","abstract_excerpt":"Given a family $\\mathcal{H}$ of graphs, we say that a graph $G$ is $\\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\\mathcal{H}$. Let $S_{t,t,t}$ be the graph obtained from $K_{1,3}$ by subdividing each edge $t-1$ times, and let $W_{t\\times t}$ be the $t$-by-$t$ hexagonal grid. Let $\\mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t \\times t}$. We prove that for every positive integer $t$ there exists $c(t)$ such that every $\\mathcal{L}_t \\cup \\{S_{t,t,t}, K_{t,t}\\}$-free $n$-vertex graph admits a tree decom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.14658","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.14658/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2501.14658","created_at":"2026-07-05T10:10:48.238726+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.14658v2","created_at":"2026-07-05T10:10:48.238726+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.14658","created_at":"2026-07-05T10:10:48.238726+00:00"},{"alias_kind":"pith_short_12","alias_value":"GEYUARCAKYXH","created_at":"2026-07-05T10:10:48.238726+00:00"},{"alias_kind":"pith_short_16","alias_value":"GEYUARCAKYXHPVSH","created_at":"2026-07-05T10:10:48.238726+00:00"},{"alias_kind":"pith_short_8","alias_value":"GEYUARCA","created_at":"2026-07-05T10:10:48.238726+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.14974","citing_title":"Coarse Balanced Separators in Biclique-Induced-Minor-Free Graphs","ref_index":12,"is_internal_anchor":false},{"citing_arxiv_id":"2605.01223","citing_title":"Tree-alpha and excluding finitely many graphs","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD","json":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD.json","graph_json":"https://pith.science/api/pith-number/GEYUARCAKYXHPVSHDEULGV7SHD/graph.json","events_json":"https://pith.science/api/pith-number/GEYUARCAKYXHPVSHDEULGV7SHD/events.json","paper":"https://pith.science/paper/GEYUARCA"},"agent_actions":{"view_html":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD","download_json":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD.json","view_paper":"https://pith.science/paper/GEYUARCA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.14658&json=true","fetch_graph":"https://pith.science/api/pith-number/GEYUARCAKYXHPVSHDEULGV7SHD/graph.json","fetch_events":"https://pith.science/api/pith-number/GEYUARCAKYXHPVSHDEULGV7SHD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD/action/storage_attestation","attest_author":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD/action/author_attestation","sign_citation":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD/action/citation_signature","submit_replication":"https://pith.science/pith/GEYUARCAKYXHPVSHDEULGV7SHD/action/replication_record"}},"created_at":"2026-07-05T10:10:48.238726+00:00","updated_at":"2026-07-05T10:10:48.238726+00:00"}