{"paper":{"title":"Algorithms and Hardness for Geodetic Set on Tree-like Digraphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Geodetic Set can be solved in polynomial time on ditrees.","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Florent Foucaud, Lucas Lorieau, Morteza Mohammad-Noori, Narges Ghareghani, Prafullkumar Tale, Rasa Parvini Oskuei","submitted_at":"2026-03-24T13:41:12Z","abstract_excerpt":"In the GEODETIC SET problem, an input is a (di)graph $G$ and integer $k$, and the objective is to decide whether there exists a vertex subset $S$ of size $k$ such that any vertex in $V(G)\\setminus S$ lies on a shortest (directed) path between two vertices in $S$. The problem has been studied on undirected and directed graphs from both algorithmic and graph-theoretical perspectives.\n  We focus on directed graphs and prove that GEODETIC SET admits a polynomial-time algorithm on ditrees, that is, digraphs with possible 2-cycles when the underlying undirected graph is a tree (after deleting possib"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"GEODETIC SET admits a polynomial-time algorithm on ditrees, that is, digraphs with possible 2-cycles when the underlying undirected graph is a tree (after deleting possible parallel edges).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The input digraphs satisfy the structural properties like being ditrees or having bounded feedback edge set, and the algorithms correctly compute shortest paths in these structures without hidden exponential factors.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Geodetic Set can be solved in polynomial time on ditrees and in FPT time parameterized by feedback edge set on 2-cycle-free digraphs, but is NP-hard on DAGs with constant feedback vertex set and pathwidth.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Geodetic Set can be solved in polynomial time on ditrees.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5cc06fc7d4d2da81bb5ef206267ce150d6289f7f75da696193fdb688528a28f6"},"source":{"id":"2603.23193","kind":"arxiv","version":3},"verdict":{"id":"f56fdb80-258a-456b-be7b-d1a32ec9d634","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T00:24:52.706814Z","strongest_claim":"GEODETIC SET admits a polynomial-time algorithm on ditrees, that is, digraphs with possible 2-cycles when the underlying undirected graph is a tree (after deleting possible parallel edges).","one_line_summary":"Geodetic Set can be solved in polynomial time on ditrees and in FPT time parameterized by feedback edge set on 2-cycle-free digraphs, but is NP-hard on DAGs with constant feedback vertex set and pathwidth.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The input digraphs satisfy the structural properties like being ditrees or having bounded feedback edge set, and the algorithms correctly compute shortest paths in these structures without hidden exponential factors.","pith_extraction_headline":"Geodetic Set can be solved in polynomial time on ditrees."},"references":{"count":30,"sample":[{"doi":"","year":2022,"title":"Discrete Mathematics345(10), 112985 (2022)","work_id":"a9367b3e-a2e0-468f-bd1c-68f613ddfb87","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Discrete Applied Mathematics323, 14–27 (2022)","work_id":"187e2e69-3f7d-4ec3-b488-bf1769e5d305","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Bergougnoux, B., Defrain, O., Mc Inerney, F.: Enumerating minimal solution sets for metric graph problems. In: Proc. of the 50th Inter- national Workshop on Graph-Theoretic Concepts in Computer Scienc","work_id":"5c54454e-9823-4a4d-a11b-683b381411b7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"In: 31st International Symposium on Algorithms and Compu- tation (ISAAC 2020)","work_id":"6d663b8a-da07-4700-9b8d-4964158c0890","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"In: Proceedings of the 6th International Conference on Algo- rithms and Discrete Applied Mathematics (CALDAM 2020)","work_id":"be4b2222-2b52-4958-89d9-24d68ab6d767","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"6ae957cbb80e737bba18a0b09e12ebe53beb2dfaf284d2b1ba2c689ebe5bf957","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"}