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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 $G_1,G_2$. 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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. 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