Paper Citation Record · LEDGER

Recognizing 2-Layer and Outer $k$-Planar Graphs

As of 23 August 2026, this Paper Citation Record lists 49 of 49 outbound references and 1 inbound Pith citation observation for arXiv:2412.04042.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2412.04042 v3

Coverage vector

measured 49 of 49 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-11T22:01:56.617323Z

measured 50 of 50 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-23T06:30:58.430688+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-07T04:32:34.008685Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: pith, observed 2026-08-07T04:32:34.455450Z

Reference resolution

49 of 49 outbound references displayed

verified exact14
verified fuzzy0
unresolved34
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation ad6fb8fb-b603-4d54-92b3-12e2c7d54ec7 · outbound

This paper cites Eliminating crossings in ordered graphs.

Recognizing 2-Layer and Outer $k$-Planar Graphs Eliminating crossings in ordered graphs

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.371485Z

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Unavailable: canonical work link unavailable.

Observation b56790a3-8c63-4b0e-bcf4-e492a6a35ded · outbound

This paper cites Edge partitions of complete geometric graphs.

Recognizing 2-Layer and Outer $k$-Planar Graphs Edge partitions of complete geometric graphs

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.377648Z

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Observation 622d3db4-1fb8-4143-84da-baf8fba47df1 · outbound

This paper cites $2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth.

Recognizing 2-Layer and Outer $k$-Planar Graphs $2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth

Reference 3

Resolution: verified exact; local_arxiv, observed 2026-08-11T22:01:57.890116Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 881a2bb3-c3a5-422e-9d47-ada361cdd681 · outbound

This paper cites Brandenburg, Andreas Glei ner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber.

Recognizing 2-Layer and Outer $k$-Planar Graphs Brandenburg, Andreas Glei ner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.388540Z

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Observation 522977fb-5c7e-4558-9141-f3cceba987f0 · outbound

This paper cites Parameterized complexity of 1-planarity.

Recognizing 2-Layer and Outer $k$-Planar Graphs Parameterized complexity of 1-planarity

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.393660Z

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Observation 0535f1ba-0f07-4245-a4a9-65af55a20c6b · outbound

This paper cites Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth.

Recognizing 2-Layer and Outer $k$-Planar Graphs Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.398197Z

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Unavailable: canonical work link unavailable.

Observation d01dfb99-29ce-4c46-ad86-7632a0eb1ae0 · outbound

This paper cites Crossing reduction in circular layouts.

Recognizing 2-Layer and Outer $k$-Planar Graphs Crossing reduction in circular layouts

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.403614Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f4433717-2754-455a-ac34-6735d89d65a1 · outbound

This paper cites Parameterized algorithms for book embedding problems.

Recognizing 2-Layer and Outer $k$-Planar Graphs Parameterized algorithms for book embedding problems

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.409139Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a29a5642-67c7-48a0-b510-2dedf5545c10 · outbound

This paper cites Min- k -planar drawings of graphs.

Recognizing 2-Layer and Outer $k$-Planar Graphs Min- k -planar drawings of graphs

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.413635Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7fc457ac-cc5f-4073-86aa-8071aae93b55 · outbound

This paper cites Constrained and Ordered Level Planarity Parameterized by the Number of Levels.

Recognizing 2-Layer and Outer $k$-Planar Graphs Constrained and Ordered Level Planarity Parameterized by the Number of Levels

Reference 10

Resolution: verified exact; local_arxiv, observed 2026-08-11T22:01:57.865351Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation ab229702-d982-4553-a7d8-a8332d7e615c · outbound

This paper cites Bodlaender.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bodlaender

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.422230Z

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Observation 20cff9be-925a-4eab-94a3-916132294ab8 · outbound

This paper cites Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T

Reference 12

Resolution: verified exact; doi, observed 2026-08-11T22:01:57.000020Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation ded047b7-de62-4863-829a-d5d796c28761 · outbound

This paper cites Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Micha Pilipczuk.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, and Micha Pilipczuk

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.431458Z

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Observation 95835e6d-9fe8-400e-8f79-4a73bfe8f5b8 · outbound

This paper cites Bodlaender, Carla Groenland, Jesper Nederlof, and C \' e line M.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bodlaender, Carla Groenland, Jesper Nederlof, and C \' e line M

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.436907Z

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Observation 68a7ae50-e679-4ea6-8b0d-3563f5e5bbb8 · outbound

This paper cites Adding one edge to planar graphs makes crossing number and 1-planarity hard.

Recognizing 2-Layer and Outer $k$-Planar Graphs Adding one edge to planar graphs makes crossing number and 1-planarity hard

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.441807Z

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Observation a6e25505-3e60-4dd7-92e7-6b5c32b25ab0 · outbound

This paper cites Pascal Gollin, Kevin Hendrey, Robert Hickingbotham, Tony Huynh, Freddie Illingworth, Youri Tamitegama, Jane Tan, and David R.

Recognizing 2-Layer and Outer $k$-Planar Graphs Pascal Gollin, Kevin Hendrey, Robert Hickingbotham, Tony Huynh, Freddie Illingworth, Youri Tamitegama, Jane Tan, and David R

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.446547Z

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Observation e86709f6-fcfb-49b5-968b-f6daba4e566e · outbound

This paper cites Beyond Outerplanarity.

Recognizing 2-Layer and Outer $k$-Planar Graphs Beyond Outerplanarity

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.451692Z

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Observation 1ffd114e-c835-4eeb-8bdb-a48acdf2748b · outbound

This paper cites Fomin, ukasz Kowalik, Daniel Lokshtanov, D \' a niel Marx, Marcin Pilipczuk, Micha Pilipczuk, and Saket Saurabh.

Recognizing 2-Layer and Outer $k$-Planar Graphs Fomin, ukasz Kowalik, Daniel Lokshtanov, D \' a niel Marx, Marcin Pilipczuk, Micha Pilipczuk, and Saket Saurabh

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.457695Z

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Observation 369769d0-c423-4e08-9e1d-2c09da4b1c4e · outbound

This paper cites A survey on graph drawing beyond planarity.

Recognizing 2-Layer and Outer $k$-Planar Graphs A survey on graph drawing beyond planarity

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.462927Z

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Observation bd9126f4-eb26-4490-a2bd-96534b264225 · outbound

This paper cites A Note on the Complexity of One-Sided Crossing Minimization of Trees.

Recognizing 2-Layer and Outer $k$-Planar Graphs A Note on the Complexity of One-Sided Crossing Minimization of Trees

Reference 20

Resolution: verified exact; local_arxiv, observed 2026-08-11T22:01:57.736237Z

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No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 53484999-f9c8-4bfe-a627-3a35bf9eabf1 · outbound

This paper cites Downey and Michael R.

Recognizing 2-Layer and Outer $k$-Planar Graphs Downey and Michael R

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.474555Z

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Observation 5e0ce76b-a135-480b-a2d4-e91497ff985c · outbound

This paper cites Hardness results for approximating the bandwidth.

Recognizing 2-Layer and Outer $k$-Planar Graphs Hardness results for approximating the bandwidth

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.479144Z

Source-reported events for the cited work

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Observation e7d4782f-c6ac-4f4d-b57b-bd1b45cd8584 · outbound

This paper cites Beyond-planar graphs: Models, structures and geometric representations ( Dagstuhl seminar 24062).

Recognizing 2-Layer and Outer $k$-Planar Graphs Beyond-planar graphs: Models, structures and geometric representations ( Dagstuhl seminar 24062)

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.483743Z

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Observation abcb9767-6c8e-449d-a965-a88feab1aad4 · outbound

This paper cites On the space and circuit complexity of parameterized problems: Classes and completeness.

Recognizing 2-Layer and Outer $k$-Planar Graphs On the space and circuit complexity of parameterized problems: Classes and completeness

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.487988Z

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Unavailable: canonical work link unavailable.

Observation 1655ee76-10cd-447c-9c99-15581eb6f489 · outbound

This paper cites Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations

Reference 25

Resolution: verified exact; local_arxiv, observed 2026-08-11T22:01:57.715360Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 474bc2c3-7cf9-446b-9ce1-72618458c0c3 · outbound

This paper cites Parameterized Complexity Theory.

Recognizing 2-Layer and Outer $k$-Planar Graphs Parameterized Complexity Theory

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.498044Z

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Observation b7df8dab-a671-43dd-944a-21828c30e1df · outbound

This paper cites Garey and David S.

Recognizing 2-Layer and Outer $k$-Planar Graphs Garey and David S

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.503110Z

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Observation 109666cd-0b2d-4516-b2e1-cbbf74fde6c8 · outbound

This paper cites Garey and David S.

Recognizing 2-Layer and Outer $k$-Planar Graphs Garey and David S

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.508036Z

Source-reported events for the cited work

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Observation 6acee6e3-3a54-4299-9f5b-b2931a2f64f1 · outbound

This paper cites Bodlaender.

Recognizing 2-Layer and Outer $k$-Planar Graphs Bodlaender

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.512720Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e30ea146-06b4-441e-b083-b86e6ffba2a7 · outbound

This paper cites Computing crossing numbers in quadratic time.

Recognizing 2-Layer and Outer $k$-Planar Graphs Computing crossing numbers in quadratic time

Reference 30

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.517094Z

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Observation c9eb42a5-d437-48a7-97cc-89d073413fe6 · outbound

This paper cites A linear-time algorithm for testing outer-1-planarity.

Recognizing 2-Layer and Outer $k$-Planar Graphs A linear-time algorithm for testing outer-1-planarity

Reference 31

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.855090Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 3a2050da-9844-4bfe-9338-c6e68f66e544 · outbound

This paper cites A linear-time algorithm for testing full outer-2-planarity.

Recognizing 2-Layer and Outer $k$-Planar Graphs A linear-time algorithm for testing full outer-2-planarity

Reference 32

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.837229Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 3676a256-5fec-4c98-bb22-999650fb7865 · outbound

This paper cites an unresolved cited work.

Recognizing 2-Layer and Outer $k$-Planar Graphs Unresolved cited work

Reference 33

Resolution: unresolved; raw_fallback, observed 2026-08-11T22:01:57.908759Z

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No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 6cefa389-7ea8-47f7-b3fb-a4b9521671db · outbound

This paper cites Computing crossing number in linear time.

Recognizing 2-Layer and Outer $k$-Planar Graphs Computing crossing number in linear time

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.539245Z

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Observation 9610086e-9282-4d08-a5ed-96781163195c · outbound

This paper cites An improved fixed-parameter algorithm for one-page crossing minimization.

Recognizing 2-Layer and Outer $k$-Planar Graphs An improved fixed-parameter algorithm for one-page crossing minimization

Reference 35

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.819426Z

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No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 4806f1fb-6776-48da-91cf-4a0a4ac50429 · outbound

This paper cites A fast and simple subexponential fixed parameter algorithm for one-sided crossing minimization.

Recognizing 2-Layer and Outer $k$-Planar Graphs A fast and simple subexponential fixed parameter algorithm for one-sided crossing minimization

Reference 36

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.803320Z

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No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 50f50050-240f-430b-8075-7018b5376000 · outbound

This paper cites A faster fixed parameter algorithm for two-layer crossing minimization.

Recognizing 2-Layer and Outer $k$-Planar Graphs A faster fixed parameter algorithm for two-layer crossing minimization

Reference 37

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.788443Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 4bade9e8-e788-4d99-8ba7-ea50dfe43084 · outbound

This paper cites Parameterized algorithms for fixed-order book drawing with bounded number of crossings per edge.

Recognizing 2-Layer and Outer $k$-Planar Graphs Parameterized algorithms for fixed-order book drawing with bounded number of crossings per edge

Reference 38

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.774764Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation e695d115-3959-48a5-879e-551d7bba790a · outbound

This paper cites On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering.

Recognizing 2-Layer and Outer $k$-Planar Graphs On parameterized algorithms for fixed-order book thickness with respect to the pathwidth of the vertex ordering

Reference 39

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.565243Z

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Observation 4937ce1b-2bb4-4966-92d6-95df1d89e433 · outbound

This paper cites One sided crossing minimization is NP -hard for sparse graphs.

Recognizing 2-Layer and Outer $k$-Planar Graphs One sided crossing minimization is NP -hard for sparse graphs

Reference 40

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.743841Z

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No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation 25eeadd1-b978-4c07-8ebb-fd7ff87e0025 · outbound

This paper cites Graphs drawn with few crossings per edge.

Recognizing 2-Layer and Outer $k$-Planar Graphs Graphs drawn with few crossings per edge

Reference 41

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.575655Z

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Observation 542898dd-20ae-4ce6-8988-93bfdf7297cc · outbound

This paper cites On space efficiency of algorithms working on structural decompositions of graphs.

Recognizing 2-Layer and Outer $k$-Planar Graphs On space efficiency of algorithms working on structural decompositions of graphs

Reference 42

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.580611Z

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Observation e087b551-9a43-4045-9016-55d187e5fcb4 · outbound

This paper cites Purchase, Christopher Pilcher, and Beryl Plimmer.

Recognizing 2-Layer and Outer $k$-Planar Graphs Purchase, Christopher Pilcher, and Beryl Plimmer

Reference 43

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.706686Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation a2358921-5a37-4cba-99e5-812a597fee2d · outbound

This paper cites an unresolved cited work.

Recognizing 2-Layer and Outer $k$-Planar Graphs Unresolved cited work

Reference 44

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.591828Z

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Observation 4d2e79b2-993f-43e0-be10-207c0972cee7 · outbound

This paper cites The graph crossing number and its variants: A survey.

Recognizing 2-Layer and Outer $k$-Planar Graphs The graph crossing number and its variants: A survey

Reference 45

Resolution: malformed identifier; no resolver link, observed 2026-08-11T22:01:56.596681Z

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Observation d8b133c6-38f3-42a3-b40b-1798dac92571 · outbound

This paper cites Sz\' e kely, and Imrich Vrto.

Recognizing 2-Layer and Outer $k$-Planar Graphs Sz\' e kely, and Imrich Vrto

Reference 46

Resolution: verified exact; doi, observed 2026-08-11T22:01:56.663346Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.

Observation f3587af4-4e5d-4599-beef-877c82c39090 · outbound

This paper cites Methods for visual understanding of hierarchical system structures.

Recognizing 2-Layer and Outer $k$-Planar Graphs Methods for visual understanding of hierarchical system structures

Reference 47

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.606717Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a46bf031-913f-43db-b757-49e1b872f2c3 · outbound

This paper cites Urschel and Jake Wellens.

Recognizing 2-Layer and Outer $k$-Planar Graphs Urschel and Jake Wellens

Reference 48

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.611983Z

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Unavailable: canonical work link unavailable.

Observation 22daf6ef-8088-400f-a02b-a54f5a84408a · outbound

This paper cites Parameterized analysis and crossing minimization problems.

Recognizing 2-Layer and Outer $k$-Planar Graphs Parameterized analysis and crossing minimization problems

Reference 49

Resolution: unresolved; no resolver link, observed 2026-08-11T22:01:56.617323Z

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Unavailable: canonical work link unavailable.

Pith citing papers

Observation b4fc7cdf-941b-4add-95f5-0bb68358e548 · inbound

Structural Parameterizations of $k$-Planarity cites this paper.

Structural Parameterizations of $k$-Planarity Recognizing 2-Layer and Outer $k$-Planar Graphs

Reference 2025

Resolution: metadata mismatch; local_arxiv, observed 2026-08-07T04:32:34.460061Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.