Pith Number

pith:4RZHF5SG

pith:2026:4RZHF5SGAEDF4QOCYWLS25TFNN

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Bond Polytope under Vertex- and Edge-sums

Hans Raj Tiwary, Petr Kolman

The bond polytope of a 1-sum or 2-sum graph is obtained directly from the bond polytopes of its component graphs.

arxiv:2601.11119 v2 · 2026-01-16 · math.CO · cs.DM · math.OC

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

25 extracted · 25 resolved · 0 Pith anchors

[1] M. Aprile and S. Fiorini. Regular matroids have polynomial extension complexity.Math. Oper. Res., 47(1):540–559, 2022 2022

[2] E. Balas. Disjunctive programming: Properties of the convex hull of feasible points.Discret. Appl. Math., 89(1-3):3–44, 1998 1998

[3] F. Barahona and A. R. Mahjoub. On the cut polytope.Math. Program., 36(2):157–173, 1986 1986

[4] J. L. Bentley. Programming pearls: algorithm design techniques.Communications of The ACM, 27:865–873, 1984 1984

[5] 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 2013

Receipt and verification

First computed	2026-05-20T01:05:06.049296Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e47272f64601065e41c2c5972d76656b6f59064b1bbb5fadcea88bf683de2f96

Aliases

arxiv: 2601.11119 · arxiv_version: 2601.11119v2 · doi: 10.48550/arxiv.2601.11119 · pith_short_12: 4RZHF5SGAEDF · pith_short_16: 4RZHF5SGAEDF4QOC · pith_short_8: 4RZHF5SG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/4RZHF5SGAEDF4QOCYWLS25TFNN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: e47272f64601065e41c2c5972d76656b6f59064b1bbb5fadcea88bf683de2f96

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e1e6fe46d2908a72b4a1d26a1f93d1356cfa9ec68594b8e91a34af3f11144ef5",
    "cross_cats_sorted": [
      "cs.DM",
      "math.OC"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-01-16T09:26:38Z",
    "title_canon_sha256": "f6a4bb11adcc89a8e6dd2740cdae6a97a53f114011bf83d5319b31d086c6b98a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.11119",
    "kind": "arxiv",
    "version": 2
  }
}