{"paper":{"title":"Bond Polytope under Vertex- and Edge-sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The bond polytope of a 1-sum or 2-sum graph is obtained directly from the bond polytopes of its component graphs.","cross_cats":["cs.DM","math.OC"],"primary_cat":"math.CO","authors_text":"Hans Raj Tiwary, Petr Kolman","submitted_at":"2026-01-16T09:26:38Z","abstract_excerpt":"A cut in a graph $G$ is called a {\\em bond} if both parts of the cut induce connected subgraphs in $G$, and the {\\em bond polytope} is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on $(K_5 \\setminus e)$-minor-free graphs, and in more general, on graphs of bounded treewidth, essentially due to clique-sum decomposition into simpler graphs.\n  We show how to obtain the bond polytope of graphs that are $1$- or $2$-sum of graphs $G_1$ and $ G_2$ from the bond polytopes of $G_1,G_2$. Using"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show how to obtain the bond polytope of graphs that are 1- or 2-sum of graphs G1 and G2 from the bond polytopes of G1,G2. Using this we show that the extension complexity of the bond polytope of (K5 minus e)-minor-free graphs is linear.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the bond polytope of the summed graph is exactly obtainable from the polytopes of G1 and G2 via the described combination rules for 1-sums and 2-sums, without extra facets or vertices arising from the identification.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Bond polytopes of 1- and 2-sums of graphs can be built from those of the summands, giving linear extension complexity for (K5 minus e)-minor-free graphs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The bond polytope of a 1-sum or 2-sum graph is obtained directly from the bond polytopes of its component graphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a7995394f927d940d39bf7bba53cb192198d38c3deb6c85b28edbca4fae41134"},"source":{"id":"2601.11119","kind":"arxiv","version":2},"verdict":{"id":"bbd87635-7830-4374-b900-07813d01f3dd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:55:41.705171Z","strongest_claim":"We show how to obtain the bond polytope of graphs that are 1- or 2-sum of graphs G1 and G2 from the bond polytopes of G1,G2. Using this we show that the extension complexity of the bond polytope of (K5 minus e)-minor-free graphs is linear.","one_line_summary":"Bond polytopes of 1- and 2-sums of graphs can be built from those of the summands, giving linear extension complexity for (K5 minus e)-minor-free graphs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the bond polytope of the summed graph is exactly obtainable from the polytopes of G1 and G2 via the described combination rules for 1-sums and 2-sums, without extra facets or vertices arising from the identification.","pith_extraction_headline":"The bond polytope of a 1-sum or 2-sum graph is obtained directly from the bond polytopes of its component graphs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.11119/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":25,"sample":[{"doi":"","year":2022,"title":"M. Aprile and S. Fiorini. Regular matroids have polynomial extension complexity.Math. Oper. Res., 47(1):540–559, 2022","work_id":"61e777ed-0a74-4e1d-b763-b9a0a29a57ca","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"E. Balas. Disjunctive programming: Properties of the convex hull of feasible points.Discret. Appl. Math., 89(1-3):3–44, 1998","work_id":"bde5af78-a8b9-4614-9d0a-12a5279796c4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1986,"title":"F. Barahona and A. R. Mahjoub. On the cut polytope.Math. Program., 36(2):157–173, 1986","work_id":"2adfdded-4221-460b-b242-f0769ad59a21","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"J. L. Bentley. Programming pearls: algorithm design techniques.Communications of The ACM, 27:865–873, 1984","work_id":"eae8782b-f6a0-41ab-b11a-32004e1abb30","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"R. Carvajal, M. Constantino, M. Goycoolea, J. P. Vielma, and A. Weintraub. Imposing connectivity constraints in forest planning models.Oper. Res., 61(4):824–836, 2013","work_id":"9ff07136-05d0-42e0-96d9-cb2cd113d15b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"c7e51f455f9823bd718f7ebfaf49828ce83a1f9ca09bc806695db6f61a036841","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"}