{"paper":{"title":"The Gallai Vertex Problem is $\\Theta_2^p$-Complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Deciding whether a graph has a vertex on all its longest paths is complete for the class Θ₂^p.","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Amir Nikabadi, Eva Rotenberg, Lasse Wulf","submitted_at":"2026-05-13T13:14:10Z","abstract_excerpt":"When a graph $G$ admits a vertex $v$ that is contained in all its longest paths, we call $v$ a Gallai vertex. These are named after Gallai, who in 1966 asked the question if it is true that every connected graph contains such a vertex. This was soon answered in the negative by Walther and Zamfirescu, who presented a graph in which every vertex is omitted by some longest path of the graph.\n  In spite of its long history, the Gallai Vertex Problem, i.e. determining whether a graph has a Gallai vertex, was until now neither known to be NP- nor co-NP-hard. In this work, we show something much stro"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the Gallai Vertex Problem is complete for the complexity class Θ₂^p = P^{NP[log n]}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polynomial-time reduction from a known Θ₂^p-complete problem to the Gallai vertex problem correctly preserves yes/no instances and runs in polynomial time.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Gallai vertex problem is Θ₂^p-complete.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Deciding whether a graph has a vertex on all its longest paths is complete for the class Θ₂^p.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"17f737627a34c8dd620236132fb204a2365fee14b24c61dcf33fa78f455c8b1c"},"source":{"id":"2605.13488","kind":"arxiv","version":1},"verdict":{"id":"d779d531-0376-48e6-8550-9b0a7bd04b87","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:21:15.868585Z","strongest_claim":"We show that the Gallai Vertex Problem is complete for the complexity class Θ₂^p = P^{NP[log n]}.","one_line_summary":"The Gallai vertex problem is Θ₂^p-complete.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The polynomial-time reduction from a known Θ₂^p-complete problem to the Gallai vertex problem correctly preserves yes/no instances and runs in polynomial time.","pith_extraction_headline":"Deciding whether a graph has a vertex on all its longest paths is complete for the class Θ₂^p."},"references":{"count":55,"sample":[{"doi":"","year":2021,"title":"Finding large","work_id":"ead5c1a9-e846-4f32-86f2-b75feb0237ca","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Finding large","work_id":"b880cb75-600f-4cb6-8ef1-13acef94c734","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.dam.2019.03.022","year":2020,"title":"2020 , issn =","work_id":"9b41baca-1571-4e58-9b70-5ae8407b3a96","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"2020 , doi =","work_id":"2d34a4d4-90e9-4fae-8286-b03e092e76c7","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.disc.2022.112985","year":2022,"title":"2022 , issn =","work_id":"fca620cc-32c0-48a3-acb1-8e494fab7072","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":55,"snapshot_sha256":"98d27b2f4b602b0cf1b332fc433169561ac427c8fd4073351f83762152ed95a0","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"}