Pith Number

pith:REIP5GIC

pith:2026:REIP5GICMKLKKPWN5R27KWPPF5

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Modelling Network Resilience: The Complexity of Some Graph Division Games

Alexander Wolff, Antonio Lauerbach, Grzegorz Gutowski, Konstanty Junosza-Szaniawski

Deciding optimal controller placement to maximize reachable surviving vertices after deletions is NP-complete or ΣP2-complete.

arxiv:2605.17572 v1 · 2026-05-17 · cs.CC · cs.GT

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Claims

C1strongest claim

We show that these natural problems are NP-complete or ΣP2-complete, depending on the specific variant. These hardness results provide limitations for optimal controller placement algorithms under different notions of quality of a solution.

C2weakest assumption

The game score is defined as the total number of surviving vertices reachable from any remaining controller under standard undirected-graph connectivity; if this reachability model does not capture the intended resilience notion, the hardness claims do not apply to the motivating application.

C3one line summary

Proves NP- and ΣP2-completeness for variants of a graph game modeling defender controller placement against vertex attacks, with polynomial algorithms for interval graphs and bounded treewidth.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] Cambridge University Press, 2009.doi:10.1017/CBO9780511804090 2009 · doi:10.1017/cbo9780511804090

[2] The vertex separation number of a graph equals its path-width 1992 · doi:10.1016/0020-0190(92

[3] Fomin and Lukasz Kowalik and Daniel Lokshtanov and D 2020 · doi:10.1007/978-3-319-21275-3

[4] 31 Ayumi Igarashi and Dominik Peters 2020 · doi:10.1109/access.2020.3047210

[5] 39 Michał Pióro, Mariusz Mycek, Artur Tomaszewski, Konstanty Junosza-Szaniawski, and Dariusz Nogalski 2003 · doi:10.1109/rndm59149.2023.10293051

Formal links

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Receipt and verification

First computed	2026-05-20T00:04:46.582474Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8910fe99026296a53ecdec75f559ef2f598285a35cf2520e73c4456c0526ffe9

Aliases

arxiv: 2605.17572 · arxiv_version: 2605.17572v1 · doi: 10.48550/arxiv.2605.17572 · pith_short_12: REIP5GICMKLK · pith_short_16: REIP5GICMKLKKPWN · pith_short_8: REIP5GIC

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/REIP5GICMKLKKPWN5R27KWPPF5 \
  | 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: 8910fe99026296a53ecdec75f559ef2f598285a35cf2520e73c4456c0526ffe9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7470d8b417a850cc792f4d7861be8885b82df9b67ea1a2203b5113e273367f59",
    "cross_cats_sorted": [
      "cs.GT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.CC",
    "submitted_at": "2026-05-17T18:00:49Z",
    "title_canon_sha256": "464d00c58ffc0ec4fd6c93cd7347118afe371060ab33965d513f5196557559d9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17572",
    "kind": "arxiv",
    "version": 1
  }
}