Paper Citation Record · LEDGER

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

As of 11 August 2026, this Paper Citation Record lists 41 of 41 outbound references and 0 inbound Pith citation observations for arXiv:2607.10941.

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pith.paper-citation-record.v1
2607.10941 v1

Coverage vector

measured 41 of 41 reference resolution

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Source: paper_references, paper_reference_links, observed 2026-07-14T08:10:14.148558Z

measured 41 of 41 standing notices

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Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

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Reference resolution

41 of 41 outbound references displayed

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Outbound references

Observation f4220f29-20f0-4c85-92e9-1462844a8a5b · outbound

This paper cites https://warwick.ac.uk/fac/sci/maths/people/staff/daniel_kral/alglogstr/ openproblems.pdf, 2016.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes https://warwick.ac.uk/fac/sci/maths/people/staff/daniel_kral/alglogstr/ openproblems.pdf, 2016

Reference 1

Resolution: unresolved; no resolver link, observed 2026-07-14T08:10:14.148558Z

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Observation 347eb531-9826-4f20-91ef-c491c32a9d26 · outbound

This paper cites Interpreting nowhere dense graph classes as a classical notion of model theory.European Journal of Combinatorics, 36:322–330, 2014.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Interpreting nowhere dense graph classes as a classical notion of model theory.European Journal of Combinatorics, 36:322–330, 2014

Reference 2

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Observation 741ee101-3339-4323-9adf-78e9796c5526 · outbound

This paper cites Implicit representation of sparse hereditary families.Discrete & Computational Geometry, 72(2):476–482, July 2023.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Implicit representation of sparse hereditary families.Discrete & Computational Geometry, 72(2):476–482, July 2023

Reference 3

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Observation 73f4ee18-bca1-477c-ad2c-94fbbaeada38 · outbound

This paper cites Second-order quantifiers and the complexity of theories.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Second-order quantifiers and the complexity of theories

Reference 4

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Observation f8ed278f-1064-4d17-a858-78f29f54266e · outbound

This paper cites $\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes $\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs

Reference 5

Resolution: unresolved; no resolver link, observed 2026-07-14T08:10:14.148558Z

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Observation c70a2d99-29ed-417a-b150-b2105280d5c3 · outbound

This paper cites Adjacency labeling schemes for small classes.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Adjacency labeling schemes for small classes

Reference 6

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Observation 0f0625fe-59f6-4295-bff7-17c2ddaa417e · outbound

This paper cites Twin-width III: Max Independent Set, Min Dominating Set, and Coloring.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Twin-width III: Max Independent Set, Min Dominating Set, and Coloring

Reference 7

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Observation 51a3f937-9cf0-49c9-9cb6-b763bb7a25a5 · outbound

This paper cites Twin-width IV: Ordered graphs and matrices.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Twin-width IV: Ordered graphs and matrices

Reference 8

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Observation 813dcb4d-aa88-426a-9b51-4c62beccf9c1 · outbound

This paper cites Twin-width I: tractable FO model checking.Journal of the ACM, 69(1):1–46, 2021.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Twin-width I: tractable FO model checking.Journal of the ACM, 69(1):1–46, 2021

Reference 9

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Observation 0ba8091f-eb8e-47bb-8c07-defb46bcfe5c · outbound

This paper cites Quasi-optimal range searching in spaces of finite VC-dimension.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Quasi-optimal range searching in spaces of finite VC-dimension

Reference 10

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Observation beceb744-d9c0-4cf4-9b5d-0164515b8edb · outbound

This paper cites Linear time solvable optimization problems on graphs of bounded clique-width.Theory of Computing Systems, 33(2):125–150, 2000.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Linear time solvable optimization problems on graphs of bounded clique-width.Theory of Computing Systems, 33(2):125–150, 2000

Reference 11

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Observation ff9a7faf-0d91-475f-817e-900ea0d305c9 · outbound

This paper cites First-order model checking on monadically stable graph classes.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes First-order model checking on monadically stable graph classes

Reference 12

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Observation e4c7f4d9-9383-4314-b127-932ec6965710 · outbound

This paper cites Near-linear time computation of Welzl orders on graphs with linear neighborhood complexity.Arxiv preprint 2602.14625, 2026.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Near-linear time computation of Welzl orders on graphs with linear neighborhood complexity.Arxiv preprint 2602.14625, 2026

Reference 13

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Observation a351dd9a-384b-4b83-82c1-ff3b0d7a8bde · outbound

This paper cites First-order model checking on struc- turally sparse graph classes.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes First-order model checking on struc- turally sparse graph classes

Reference 14

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Observation 783c8b04-75f6-4f08-ba5d-ff2136843d5c · outbound

This paper cites Flip-breakability: A combinatorial dichotomy for monadically dependent graph classes.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Flip-breakability: A combinatorial dichotomy for monadically dependent graph classes

Reference 15

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Observation 03c20566-65c9-4f4f-b8fd-1e2a4006dc04 · outbound

This paper cites Merge-width and first-order model checking.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Merge-width and first-order model checking

Reference 16

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Observation d1811945-b4c3-446c-a0f9-fe3d547200e8 · outbound

This paper cites Better diameter algorithms for bounded VC-dimension graphs and geometric intersection graphs.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Better diameter algorithms for bounded VC-dimension graphs and geometric intersection graphs

Reference 17

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Observation 255f1334-e514-45d0-a4c8-2c8a6d186118 · outbound

This paper cites Induced subdivisions and bounded expansion.European Journal of Combina- torics, 69:143–148, 2018.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Induced subdivisions and bounded expansion.European Journal of Combina- torics, 69:143–148, 2018

Reference 18

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Observation fdb4337b-1c47-4173-a8f9-5264ddc4ab29 · outbound

This paper cites Testing first-order properties for subclasses of sparse graphs.J.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Testing first-order properties for subclasses of sparse graphs.J

Reference 19

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Observation 77c86fd8-6557-453e-8490-c34d3046b83a · outbound

This paper cites Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michał Pilipczuk, Roman Rabinovich, and Sebastian Siebertz.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michał Pilipczuk, Roman Rabinovich, and Sebastian Siebertz

Reference 20

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Observation 99fff0e5-c100-4f75-8462-388fe3f8d7ab · outbound

This paper cites an unresolved cited work.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Unresolved cited work

Reference 21

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Observation e5eb75ed-b4c6-4268-95f7-22ca4792810b · outbound

This paper cites Deciding first-order properties of nowhere dense graphs.Journal of the ACM, 64(3):1–32, 2017.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Deciding first-order properties of nowhere dense graphs.Journal of the ACM, 64(3):1–32, 2017

Reference 22

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Observation 578b0c02-e6ce-4656-913a-4179e5816a2d · outbound

This paper cites Haussler, N.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Haussler, N

Reference 23

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Observation 46c201ad-8cbc-4386-a49a-5a76c6a47a9a · outbound

This paper cites Sphere packing numbers for subsets of the Booleann-cube with bounded Vapnik-Chervonenkis dimension.Journal of Combinatorial Theory, Series A, 69(2):217–232, 1995.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Sphere packing numbers for subsets of the Booleann-cube with bounded Vapnik-Chervonenkis dimension.Journal of Combinatorial Theory, Series A, 69(2):217–232, 1995

Reference 24

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Observation b15c07ec-65a3-418e-9c75-a94c121bf870 · outbound

This paper cites PhD thesis, University of Bremen, 2024.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes PhD thesis, University of Bremen, 2024

Reference 25

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Observation 46e88ea5-a402-4f04-8337-a62a574f4220 · outbound

This paper cites Springer Berlin Heidelberg, 1999.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Springer Berlin Heidelberg, 1999

Reference 26

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Observation ce1dddfa-cdd8-419e-905a-c9e8f6fd5612 · outbound

This paper cites Rankwidth meets stability.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Rankwidth meets stability

Reference 27

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Observation df12813a-1959-4397-a2cf-fcfb93ffce99 · outbound

This paper cites Graph spanners.Journal of Graph Theory, 13(1):99–116, 1989.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Graph spanners.Journal of Graph Theory, 13(1):99–116, 1989

Reference 28

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Observation 39f81c04-524b-461f-a110-72fa34e47492 · outbound

This paper cites Chapter 1: Measuring sparsity.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Chapter 1: Measuring sparsity

Reference 29

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Observation 7bd0d798-8810-4086-8267-d0601920df42 · outbound

This paper cites Flipping and Forking.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Flipping and Forking

Reference 30

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Observation 65262cb1-c468-4414-8cac-439e407f8dee · outbound

This paper cites On the density of families of sets.Journal of Combinatorial Theory, Series A, 13(1):145–147, 1972.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes On the density of families of sets.Journal of Combinatorial Theory, Series A, 13(1):145–147, 1972

Reference 31

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Observation 691490ee-22a2-4b7b-86e0-3917db7e1a75 · outbound

This paper cites A combinatorial problem; stability and order for models and theories in infinitary languages.Pacific Journal of Mathematics, 41(1):247–261, 1972.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes A combinatorial problem; stability and order for models and theories in infinitary languages.Pacific Journal of Mathematics, 41(1):247–261, 1972

Reference 32

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Observation 03dbd727-03d6-4cd3-9354-718c1208d7bb · outbound

This paper cites Flip-width: Cops and Robber on dense graphs.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Flip-width: Cops and Robber on dense graphs

Reference 33

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Observation 24690afc-1034-4ea2-bd08-0d077631d95f · outbound

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Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Unresolved cited work

Reference 34

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Observation 0f699751-191f-44ff-abd9-b1353085c92a · outbound

This paper cites Fast shortest path in graphs with sparse signed tree models and applications.Arxiv preprint 2602.16605, 2026.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Fast shortest path in graphs with sparse signed tree models and applications.Arxiv preprint 2602.16605, 2026

Reference 35

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Observation 556f04af-4d58-4440-af29-5d425b198a03 · outbound

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Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Unresolved cited work

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Observation 86a31a73-fb69-4161-898d-8098cabdb30b · outbound

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Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Unresolved cited work

Reference 37

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Observation 6f24cd22-7ba0-4c3b-bc67-3f66778fcd14 · outbound

This paper cites Combining these three points yields a terminalk-sparsification withk⩽dand sizessatisfying |B| polylogd(|A|) ⩽s⩽|A| ·subpoly B,d(|A|).

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Combining these three points yields a terminalk-sparsification withk⩽dand sizessatisfying |B| polylogd(|A|) ⩽s⩽|A| ·subpoly B,d(|A|)

Reference 38

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Observation c06fb4cc-5c6b-4516-88d2-cb266862dad9 · outbound

This paper cites Thus, the new sparsification has dimension at most d−1.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Thus, the new sparsification has dimension at most d−1

Reference 39

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Observation 9e5b65a1-eb57-4e24-8ab8-dcd086154362 · outbound

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Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Unresolved cited work

Reference 40

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Observation dfe58644-267a-40cc-8416-071d2053fe37 · outbound

This paper cites Now we can use Lemma 27 to prove the key lemma about fractional twins.

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes Now we can use Lemma 27 to prove the key lemma about fractional twins

Reference 41

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