research-article

Planar Graphs Have Bounded Queue-Number

Authors:

Vida Dujmović,

Gwenaël Joret,

Torsten Ueckerdt,

David R. WoodAuthors Info & Claims

Journal of the ACM (JACM), Volume 67, Issue 4

Article No.: 22, Pages 1 - 38

https://doi.org/10.1145/3385731

Published: 06 August 2020 Publication History

Abstract

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number.

Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

References

[1]

Martin Aigner and Günter M. Ziegler. 2010. Proofs from the Book (4th ed.). Springer.

[2]

Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. 2018. Queue layouts of planar 3-trees. In Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018) (Lecture Notes in Computer Science), Therese C. Biedl and Andreas Kerren (Eds.), vol. 11282. Springer, 213–226.

Digital Library

[3]

Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Sergey Pupyrev. 2020. Queue layouts of planar 3-trees. Algorithmica (2020).

[4]

Noga Alon and Vera Asodi. 2002. Sparse universal graphs. J. Comput. Appl. Math. 142, 1 (2002), 1–11. MR: 1910514.

Digital Library

[5]

Noga Alon and Michael Capalbo. 2007. Sparse universal graphs for bounded-degree graphs. Random Structures Algorithms 31, 2 (2007), 123–133.

Digital Library

[6]

Noga Alon, Jarosław Grytczuk, Mariusz Hałuszczak, and Oliver Riordan. 2002. Nonrepetitive colorings of graphs. Random Structures Algorithms 21, 3–4 (2002), 336–346. MR: 1945373.

Digital Library

[7]

Noga Alon, Bojan Mohar, and Daniel P. Sanders. 1996. On acyclic colorings of graphs on surfaces. Israel J. Math. 94 (1996), 273–283.

[8]

Thomas Andreae. 1986. On a pursuit game played on graphs for which a minor is excluded. J. Comb. Theory, Ser. B 41, 1 (1986), 37–47. MR: 0854602.

Digital Library

[9]

László Babai, Fan R. K. Chung, Paul Erdős, Ron L. Graham, and Joel H. Spencer. 1982. On graphs which contain all sparse graphs. In Theory and Practice of Combinatorics. North-Holland Math. Stud., Vol. 60. 21–26. MR: 806964.

[10]

Brenda S. Baker. 1994. Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 1 (1994), 153–180. MR: 1369197.

Digital Library

[11]

Michael J. Bannister, William E. Devanny, Vida Dujmović, David Eppstein, and David R. Wood. 2019. Track layouts, layered path decompositions, and leveled planarity. Algorithmica 81, 4 (2019), 1561–1583. MR: 3936168.

Digital Library

[12]

Michael A. Bekos, Henry Förster, Martin Gronemann, Tamara Mchedlidze, Fabrizio Montecchiani, Chrysanthi N. Raftopoulou, and Torsten Ueckerdt. 2019. Planar graphs of bounded degree have bounded queue number. SIAM J. Comput. 48, 5 (2019), 1487–1502.

Digital Library

[13]

Sandeep N. Bhatt, Fan R. K. Chung, Frank Thomson Leighton, and Arnold L. Rosenberg. 1989. Universal graphs for bounded-degree trees and planar graphs. SIAM J. Discrete Math. 2, 2 (1989), 145–155. MR: 990447.

Digital Library

[14]

Hans L. Bodlaender. 1998. A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209, 1-2 (1998), 1–45. MR: 1647486.

Digital Library

[15]

Hans L. Bodlaender and Joost Engelfriet. 1997. Domino treewidth. J. Algorithms 24, 1 (1997), 94–123. MR: 1453952.

Digital Library

[16]

Marthe Bonamy, Cyril Gavoille, and Michał Pilipczuk. 2020. Shorter labeling schemes for planar graphs. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA’20), Shuchi Chawla (Ed.). 446–462. arXiv: 1908.03341.

[17]

Oleg V. Borodin. 1979. On acyclic colorings of planar graphs. Discrete Math. 25, 3 (1979), 211–236.

[18]

Oleg V. Borodin, Alexandr V. Kostochka, Jaroslav Nešetřil, André Raspaud, and Éric Sopena. 1998. On universal graphs for planar oriented graphs of a given girth. Discrete Math. 188, 1–3 (1998), 73–85.

Digital Library

[19]

Julia Böttcher, Klaas Paul Pruessmann, Anusch Taraz, and Andreas Würfl. 2010. Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. European J. Combin. 31, 5 (2010), 1217–1227.

Digital Library

[20]

Jonathan F. Buss and Peter Shor. 1984. On the pagenumber of planar graphs. In Proceedings of the 16th ACM Symposium. on Theory of Computing (STOC’84). ACM, 98–100.

[21]

Sergio Cabello, Éric Colin de Verdière, and Francis Lazarus. 2012. Algorithms for the edge-width of an embedded graph. Comput. Geom. 45, 5--6 (2012), 215–224.

Digital Library

[22]

L. Sunil Chandran, Alexandr Kostochka, and J. Krishnam Raju. 2008. Hadwiger number and the Cartesian product of graphs. Graphs Combin. 24, 4 (2008), 291–301.

Digital Library

[23]

Zhi-Zhong Chen. 2001. Approximation algorithms for independent sets in map graphs. J. Algorithms 41, 1 (2001), 20–40.

Digital Library

[24]

Zhi-Zhong Chen. 2007. New bounds on the edge number of a k-map graph. J. Graph Theory 55, 4 (2007), 267–290. MR: 2336801.

Digital Library

[25]

Zhi-Zhong Chen, Michelangelo Grigni, and Christos H. Papadimitriou. 2002. Map graphs. J. ACM 49, 2 (2002), 127–138. MR: 2147819.

Digital Library

[26]

Robert F. Cohen, Peter Eades, Tao Lin, and Frank Ruskey. 1996. Three-dimensional graph drawing. Algorithmica 17, 2 (1996), 199–208.

[27]

Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. 2005. Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs. ACM Trans. Algorithms 1, 1 (2005), 33–47.

Digital Library

[28]

Erik D. Demaine and MohammadTaghi Hajiaghayi. 2004. Diameter and treewidth in minor-closed graph families, revisited. Algorithmica 40, 3 (2004), 211–215. MR: 2080518.

[29]

Erik D. Demaine and MohammadTaghi Hajiaghayi. 2004. Equivalence of local treewidth and linear local treewidth and its algorithmic applications. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’04). SIAM, 840–849. http://dl.acm.org/citation.cfm?id=982792.982919.

[30]

Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. 2005. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05). IEEE, 637–646.

Digital Library

[31]

Matt DeVos, Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, Bruce Reed, Paul Seymour, and Dirk Vertigan. 2004. Excluding any graph as a minor allows a low tree-width 2-coloring. J. Combin. Theory Ser. B 91, 1 (2004), 25–41. MR: 2047529.

Digital Library

[32]

Giuseppe Di Battista, Fabrizio Frati, and János Pach. 2013. On the queue number of planar graphs. SIAM J. Comput. 42, 6 (2013), 2243–2285. MR: 3141759.

Digital Library

[33]

Emilio Di Giacomo and Henk Meijer. 2004. Track drawings of graphs with constant queue number. In Proceedings of the 11th International Symposium on Graph Drawing (GD’03) (Lecture Notes in Computer Science), Giuseppe Liotta (Ed.), Vol. 2912. Springer, 214–225.

[34]

Reinhard Diestel. 2018. Graph Theory (5th ed.). Graduate Texts in Mathematics, Vol. 173. Springer. MR: 3822066.

[35]

Reinhard Diestel and Daniela Kühn. 2005. Graph minor hierarchies. Discrete Appl. Math. 145, 2 (2005), 167–182.

Digital Library

[36]

Guoli Ding and Bogdan Oporowski. 1995. Some results on tree decomposition of graphs. J. Graph Theory 20, 4 (1995), 481–499. MR: 1358539.

Digital Library

[37]

Guoli Ding and Bogdan Oporowski. 1996. On tree-partitions of graphs. Discrete Math. 149, 1–3 (1996), 45–58. MR: 1375097.

Digital Library

[38]

Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, and Dirk Vertigan. 1998. Partitioning graphs of bounded tree-width. Combinatorica 18, 1 (1998), 1–12. MR: 1645638.

[39]

Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, and Dirk Vertigan. 2000. Surfaces, tree-width, clique-minors, and partitions. J. Combin. Theory Ser. B 79, 2 (2000), 221–246. MR: 1769192.

Digital Library

[40]

Michał Dębski, Stefan Felsner, Piotr Micek, and Felix Schröder. 2020. Improved bounds for centered colorings. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA’20), Shuchi Chawla (Ed.). 2212–2226. arXiv: 1907.04586.

[41]

Vida Dujmović. 2015. Graph layouts via layered separators. J. Combin. Theory Series B. 110 (2015), 79–89. MR: 3279388.

Digital Library

[42]

Vida Dujmović, David Eppstein, Gwenaël Joret, Pat Morin, and David R. Wood. 2018. Minor-closed graph classes with bounded layered pathwidth. arXiv: 1810.08314.

[43]

Vida Dujmović, David Eppstein, and David R. Wood. 2017. Structure of graphs with locally restricted crossings. SIAM J. Discrete Math. 31, 2 (2017), 805–824.

[44]

Vida Dujmović, Louis Esperet, Gwenaël Joret, Cyril Gavoille, Piotr Micek, and Pat Morin. 2020. Adjacency labelling for planar graphs (and beyond). (2020). arXiv: 2003.04280.

[45]

Vida Dujmović, Louis Esperet, Gwenaël Joret, Bartosz Walczak, and David R. Wood. 2020. Planar graphs have bounded nonrepetitive chromatic number. Advances in Combinatorics 5 (2020).

[46]

Vida Dujmović, Louis Esperet, Pat Morin, Bartosz Walczak, and David R. Wood. 2020. Clustered 3-colouring graphs of bounded degree. (2020). arXiv: 2002.11721.

[47]

Vida Dujmović and Fabrizio Frati. 2018. Stack and queue layouts via layered separators. J. Graph Algorithms Appl. 22, 1 (2018), 89–99. MR: 3757347.

[48]

Vida Dujmović, Pat Morin, and David R. Wood. 2005. Layout of graphs with bounded tree-width. SIAM J. Comput. 34, 3 (2005), 553–579. MR: 2137079.

Digital Library

[49]

Vida Dujmović, Pat Morin, and David R. Wood. 2017. Layered separators in minor-closed graph classes with applications. J. Combin. Theory Ser. B 127 (2017), 111–147. arXiv:1306.1595MR: 3704658.

[50]

Vida Dujmović, Pat Morin, and David R. Wood. 2019. Queue layouts of graphs with bounded degree and bounded genus. (2019). arXiv: 1901.05594.

[51]

Vida Dujmović, Attila Pór, and David R. Wood. 2004. Track layouts of graphs. Discrete Math. Theor. Comput. Sci. 6, 2 (2004), 497–522. http://dmtcs.episciences.org/315 MR: 2180055.

[52]

Vida Dujmović and David R. Wood. 2004. On linear layouts of graphs. Discrete Math. Theor. Comput. Sci. 6, 2 (2004), 339–358. http://dmtcs.episciences.org/317 MR: 2081479.

[53]

Vida Dujmović and David R. Wood. 2004. Three-dimensional grid drawings with sub-quadratic volume. In Towards a Theory of Geometric Graphs, János Pach (Ed.). Contemporary Mathematics, vol. 342. Amer. Math. Soc., 55–66. MR: 2065252.

[54]

Vida Dujmović and David R. Wood. 2005. Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci. 7 (2005), 155–202. http://dmtcs.episciences.org/346 MR: 2164064.

[55]

Vida Dujmović and David R. Wood. 2006. Upward three-dimensional grid drawings of graphs. Order 23, 1 (2006), 1–20. MR: 2258457.

[56]

Zdeněk Dvořák, Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R. Wood. 2020. Notes on graph product structure theory. (2020). arXiv: 2001.08860.

[57]

Zdeněk Dvořák and Robin Thomas. 2014. List-coloring apex-minor-free graphs. arXiv: 1401.1399.

[58]

David Eppstein. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 3–4 (2000), 275–291. MR: 1759751.

[59]

Jeff Erickson and Kim Whittlesey. 2005. Greedy optimal homotopy and homology generators. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM, 1038–1046.

[60]

Stefan Felsner, Giussepe Liotta, and Stephen K. Wismath. 2002. Straight-line drawings on restricted integer grids in two and three dimensions. In Proceedings of the 9th International Symposium on Graph Drawing (GD’01)(Computer Science), Petra Mutzel, Michael Jünger, and Sebastian Leipert (Eds.), vol. 2265. Springer, 328–342.

[61]

Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. 2012. Bidimensionality and geometric graphs. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms. 1563–1575. MR: 3205314.

[62]

Jacob Fox and János Pach. 2010. A separator theorem for string graphs and its applications. Combin. Probab. Comput. 19, 3 (2010), 371–390.

Digital Library

[63]

Jacob Fox and János Pach. 2014. Applications of a new separator theorem for string graphs. Combin. Probab. Comput. 23, 1 (2014), 66–74.

[64]

Daniel J. Harvey and David R. Wood. 2017. Parameters tied to treewidth. J. Graph Theory 84, 4 (2017), 364–385. MR: 3623383.

[65]

Toru Hasunuma. 2007. Queue layouts of iterated line directed graphs. Discrete Appl. Math. 155, 9 (2007), 1141–1154. MR: 2321020.

Digital Library

[66]

Lenwood S. Heath, F. Thomson Leighton, and Arnold L. Rosenberg. 1992. Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math. 5, 3 (1992), 398–412. MR: 1172748.

Digital Library

[67]

Lenwood S. Heath and Sriram V. Pemmaraju. 1997. Stack and queue layouts of posets. SIAM J. Discrete Math. 10, 4 (1997), 599–625.

Digital Library

[68]

Lenwood S. Heath and Arnold L. Rosenberg. 1992. Laying out graphs using queues. SIAM J. Comput. 21, 5 (1992), 927–958. MR: 1181408.

Digital Library

[69]

Lenwood S. Heath and Arnold L. Rosenberg. 2011. Graph Layout using Queues. (2011). https://www.researchgate.net/publication/220616637_Laying_Out_Graphs_Using_Queues

[70]

Percy J. Heawood. 1890. Map colour theorem. Quart. J. Pure Appl. Math. 24 (1890), 332–338.

[71]

Jan van den Heuvel, Patrice Ossona de Mendez, Daniel Quiroz, Roman Rabinovich, and Sebastian Siebertz. 2017. On the generalised colouring numbers of graphs that exclude a fixed minor. European J. Combin. 66 (2017), 129–144.

[72]

Jan van den Heuvel and David R. Wood. 2017. Improper colourings inspired by Hadwiger’s Conjecture. (2017). arXiv: 1704.06536.

[73]

Jan van den Heuvel and David R. Wood. 2018. Improper colourings inspired by Hadwiger’s conjecture. J. London Math. Soc. 98 (2018), 129–148. Issue 1.

[74]

Sampath Kannan, Moni Naor, and Steven Rudich. 1988. Implicit representation of graphs. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC 1988). 334–343.

Digital Library

[75]

Sampath Kannan, Moni Naor, and Steven Rudich. 1992. Implicit representation of graphs. SIAM J. Discrete Math. 5, 4 (1992), 596–603.

Digital Library

[76]

Hal A. Kierstead and Daqing Yang. 2003. Orderings on graphs and game coloring number. Order 20, 3 (2003), 255–264.

[77]

Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. 2018. The queue-number of posets of bounded width or height. In Graph Drawing and Network Visualization (Lecture Notes in Computer Science), vol. 11282. Springer, 200–212.

[78]

Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. 2017. An annotated bibliography on 1-planarity. Comput. Sci. Rev. 25 (2017), 49–67. MR: 3697129.

[79]

Andreĭ Kotlov. 2001. Minors and strong products. European J. Combin. 22, 4 (2001), 511–512. MR: 1829745.

Digital Library

[80]

Kyohei Kozawa, Yota Otachi, and Koichi Yamazaki. 2014. Lower bounds for treewidth of product graphs. Discrete Appl. Math. 162 (2014), 251–258. MR: 3128527.

Digital Library

[81]

Jan Kratochvíl. 1991. String graphs. II. Recognizing string graphs is NP-hard. J. Combin. Theory Ser. B 52, 1 (1991), 67–78.

Digital Library

[82]

Jan Kratochvíl and Michal Vaner. 2012. A note on planar partial 3-trees. (2012). arXiv: 1210.8113.

[83]

Chun-Hung Liu and David R. Wood. 2019. Clustered graph coloring and layered treewidth. arXiv: 1905.08969.

[84]

Seth M. Malitz. 1994. Genus g graphs have pagenumber O( √g) . J. Algorithms 17, 1 (1994), 85–109. MR: 1279270.

Digital Library

[85]

Bojan Mohar. 1999. A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math. 12, 1 (1999), 6–26.

Digital Library

[86]

Bojan Mohar and Carsten Thomassen. 2001. Graphs on Surfaces. Johns Hopkins University Press. MR: 1844449.

[87]

Pat Morin. 2020. A fast algorithm for the product structure of planar graphs. (2020). arXiv: 2004.02530.

[88]

Jaroslav Nešetřil and Patrice Ossona de Mendez. 2012. Sparsity. Algorithms and Combinatorics, vol. 28. Springer. MR: 2920058.

[89]

L. Taylor Ollmann. 1973. On the book thicknesses of various graphs. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing (Congr. Numer.), Frederick Hoffman, Roy B. Levow, and Robert S. D. Thomas (Eds.), vol. VIII. Utilitas Math., 459.

[90]

Patrice Ossona de Mendez, Sang-il Oum, and David R. Wood. 2019. Defective colouring of graphs excluding a subgraph or minor. Combinatorica 39, 2 (2019), 377–410.

Digital Library

[91]

János Pach, Torsten Thiele, and Géza Tóth. 1999. Three-dimensional grid drawings of graphs. In Advances in Discrete and Computational Geometry, Bernard Chazelle, Jacob E. Goodman, and Richard Pollack (Eds.). Contemporary Mathematics, vol. 223. Amer. Math. Soc., 251–255. MR: 1661387.

[92]

János Pach and Géza Tóth. 2002. Recognizing string graphs is decidable. Discrete Comput. Geom. 28, 4 (2002), 593–606.

Digital Library

[93]

Sriram V. Pemmaraju. 1992. Exploring the Powers of Stacks and Queues via Graph Layouts. Ph.D. Dissertation. Virginia Polytechnic Institute and State University.

[94]

Michał Pilipczuk and Sebastian Siebertz. 2019. Polynomial bounds for centered colorings on proper minor-closed graph classes. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, Timothy M. Chan (Ed.). 1501–1520. arXiv: 1807.03683.

[95]

Sergey Pupyrev. 2019. Improved bounds for track numbers of planar graphs. (2019). arXiv: 1910.14153.

[96]

Bruce A. Reed. 1997. Tree width and tangles: A new connectivity measure and some applications. In Surveys in Combinatorics. London Math. Soc. Lecture Note Ser., vol. 241. Cambridge University Press, 87–162. MR: 1477746.

[97]

Bruce A. Reed and Paul Seymour. 1998. Fractional colouring and Hadwiger’s conjecture. J. Combin. Theory Ser. B 74, 2 (1998), 147–152. MR: 1654153.

Digital Library

[98]

S. Rengarajan and C. E. Veni Madhavan. 1995. Stack and queue number of 2-trees. In Proceedings of the 1st Annual International Conference on Computing and Combinatorics (COCOON’95) (Lecture Notes in Computer Science), Ding-Zhu Du and Ming Li (Eds.), vol. 959. Springer, 203–212.

[99]

Neil Robertson and Paul Seymour. 1986. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 3 (1986), 309–322. MR: 0855559.

[100]

Neil Robertson and Paul Seymour. 2003. Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B 89, 1 (2003), 43–76. MR: 1999736.

Digital Library

[101]

Marcus Schaefer, Eric Sedgwick, and Daniel Štefankovič. 2003. Recognizing string graphs in NP. J. Comput. System Sci. 67, 2 (2003), 365–380.

Digital Library

[102]

Marcus Schaefer and Daniel Štefankovič. 2004. Decidability of string graphs. J. Comput. System Sci. 68, 2 (2004), 319–334.

Digital Library

[103]

Alex Scott, Paul Seymour, and David R. Wood. 2019. Bad news for chordal partitions. J. Graph Theory 90 (2019), 5–12.

[104]

Detlef Seese. 1985. Tree-partite graphs and the complexity of algorithms. In Proceedings of the International Conference on Fundamentals of Computation Theory (Lecture Notes in Computer Science), Lothar Budach (Ed.), vol. 199. Springer, 412–421. MR: 0821258.

[105]

Farhad Shahrokhi. 2013. New representation results for planar graphs. In Proceedings of the 29th European Workshop on Computational Geometry (EuroCG 2013). 177–180. arXiv: 1502.06175.

[106]

Jiun-Jie Wang. 2017. Layouts for plane graphs on constant number of tracks. (2017). arXiv: 1708.02114.

[107]

Veit Wiechert. 2017. On the queue-number of graphs with bounded tree-width. Electron. J. Combin. 24, 1 (2017), 1.65. https://doi.org/10.37236/6429 MR: 3651947.

[108]

David R. Wood. 2005. Queue layouts of graph products and powers. Discrete Math. Theor. Comput. Sci. 7, 1 (2005), 255–268. http://dmtcs.episciences.org/352 MR: 2183176.

[109]

David R. Wood. 2008. Bounded-degree graphs have arbitrarily large queue-number. Discrete Math. Theor. Comput. Sci. 10, 1 (2008), 27–34. http://dmtcs.episciences.org/434 MR: 2369152.

[110]

David R. Wood. 2008. The structure of Cartesian products. (2008). https://www.birs.ca/workshops/2008/08w5079/report08w5079.pdf

[111]

David R. Wood. 2009. On tree-partition-width. European J. Combin. 30, 5 (2009), 1245–1253. MR: 2514645.

Digital Library

[112]

David R. Wood. 2011. Clique minors in Cartesian products of graphs. New York J. Math. 17 (2011), 627–682. http://nyjm.albany.edu/j/2011/17-28.html

[113]

David R. Wood. 2013. Treewidth of Cartesian products of highly connected graphs. J. Graph Theory 73, 3 (2013), 318–321.

[114]

Zefang Wu, Xu Yang, and Qinglin Yu. 2010. A note on graph minors and strong products. Appl. Math. Lett. 23, 10 (2010), 1179–1182. MR: 2665591.

[115]

Mihalis Yannakakis. 1989. Embedding planar graphs in four pages. J. Comput. System Sci. 38, 1 (1989), 36–67. MR: 0990049.

Digital Library

[116]

Vida Dujmović, Pat Morin, and David R. Wood. 2019. Graph product structure for non-minor-closed classes. arXiv: 1907.05168.

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Index Terms

Planar Graphs Have Bounded Queue-Number
1. Mathematics of computing

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Journal of the ACM Volume 67, Issue 4

August 2020

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DOI:10.1145/3403612

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Published: 06 August 2020

Online AM: 07 May 2020

Accepted: 01 February 2020

Received: 01 May 2019

Published in JACM Volume 67, Issue 4

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Polish National Science Center
Australian Research Council
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Fernau HFoucaud FMann KPadariya URao K.N. R(2025)Parameterizing path partitionsTheoretical Computer Science10.1016/j.tcs.2024.1150291028(115029)Online publication date: Feb-2025
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Cortés PKumar PMoore BOssona de Mendez PQuiroz D(2025)Subchromatic numbers of powers of graphs with excluded minorsDiscrete Mathematics10.1016/j.disc.2024.114377348:4(114377)Online publication date: Apr-2025
https://doi.org/10.1016/j.disc.2024.114377
Distel MHickingbotham RSeweryn MWood D(2024)Powers of planar graphs, product structure, and blocking partitionsInnovations in Graph Theory10.5802/igt.41(39-86)Online publication date: 26-Nov-2024
https://doi.org/10.5802/igt.4
Distel MDujmović VEppstein DHickingbotham RJoret GMicek PMorin PSeweryn MWood D(2024)Product Structure Extension of the Alon–Seymour–Thomas TheoremSIAM Journal on Discrete Mathematics10.1137/23M159177338:3(2095-2107)Online publication date: 9-Jul-2024
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Hickingbotham RWood D(2024)Shallow Minors, Graph Products, and Beyond-Planar GraphsSIAM Journal on Discrete Mathematics10.1137/22M154029638:1(1057-1089)Online publication date: 13-Mar-2024
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Illingworth FScott AWood D(2024)Product structure of graphs with an excluded minorTransactions of the American Mathematical Society, Series B10.1090/btran/19211:35(1233-1248)Online publication date: 1-Nov-2024
https://doi.org/10.1090/btran/192
Hliněný PStraka A(2024)Stack and queue numbers of graphs revisitedEuropean Journal of Combinatorics10.1016/j.ejc.2024.104094(104094)Online publication date: Dec-2024
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Esperet LWood D(2024)Colouring strong productsEuropean Journal of Combinatorics10.1016/j.ejc.2023.103847121(103847)Online publication date: Oct-2024
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Joret GRambaud C(2024)Neighborhood Complexity of Planar GraphsCombinatorica10.1007/s00493-024-00110-644:5(1115-1148)Online publication date: 1-Oct-2024
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Angelini PBekos MDa Lozzo GGronemann MMontecchiani FTappini A(2024)Recognizing Map Graphs of Bounded TreewidthAlgorithmica10.1007/s00453-023-01180-686:2(613-637)Online publication date: 1-Feb-2024
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