Pith Number

pith:3NQQBV4Y

pith:2026:3NQQBV4YASQSCZP5HVKTYQKEZU

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Two Results on Outer-String Graphs

Jan Kratochv\'il, Peter Stumpf, Todor Anti\'c, V\'it Jel\'inek

Bipartite graphs with a given cyclic order admit a polynomial-time test for constrained outer-string representations based on a single forbidden configuration, while recognizing outer-k-string representations is NP-hard for any fixed k.

arxiv:2605.12253 v2 · 2026-05-12 · math.CO · cs.CG

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3 Author claim open · sign in to claim

4 Citations open

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Claims

C1strongest claim

For a bipartite graph G (and more generally for any {C3,C5}-free graph G) with a given cyclic order of vertices, we can decide in polynomial time whether G admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration.

C2weakest assumption

The algorithm's correctness rests on the claim that a single forbidden configuration completely characterizes the graphs that admit constrained outer-string representations (abstract, paragraph describing the first result and reference to Biedl et al. [GD 2024]).

C3one line summary

Polynomial-time recognition for constrained outer-string representations of {C3,C5}-free graphs with cyclic order, plus NP-hardness for outer-k-string recognition for any fixed k.

References

31 extracted · 31 resolved · 0 Pith anchors

[1] Iqbal Hossain and Stephen G 2017

[2] Cyclic ordering is 1977 · doi:10.1016/0304-3975(77)90005-6

[3] V. On Grounded. Electron. J. Comb. , volume =. 2019 , OPTurl = 2019

[4] Constrained Outer-String Representations , booktitle = 2024

[5] Constrained Outer-String Representations , journal = 2026

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:00:42.970712Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

db6100d79804a12165fd3d553c4144cd1bf621e1d914ae440249c5fcf6341aba

Aliases

arxiv: 2605.12253 · arxiv_version: 2605.12253v2 · doi: 10.48550/arxiv.2605.12253 · pith_short_12: 3NQQBV4YASQS · pith_short_16: 3NQQBV4YASQSCZP5 · pith_short_8: 3NQQBV4Y

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3NQQBV4YASQSCZP5HVKTYQKEZU \
  | 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: db6100d79804a12165fd3d553c4144cd1bf621e1d914ae440249c5fcf6341aba

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3f879db7e43e31f2ca9878fef97f7eaebdb8be85ad64ee1dfa3d6d3c6f47169b",
    "cross_cats_sorted": [
      "cs.CG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-12T15:22:54Z",
    "title_canon_sha256": "f2704350d1d517bfccec8491a41000aefe10c2b182d8cea8028ed7ace4918674"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12253",
    "kind": "arxiv",
    "version": 2
  }
}