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### Planar Graphs of Bounded Degree have Constant Queue Number [article]

Michael A. Bekos, Henry Förster, Martin Gronemann, Tamara Mchedlidze, Fabrizio Montecchiani, Chrysanthi Raftopoulou, Torsten Ueckerdt
2019 arXiv   pre-print
We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number.  ...  edge) of bounded degree has bounded queue number.  ...  Wood for pointing out an issue in an earlier version of this paper and the anonymous referees of both the journal and the conference version of this paper for their valuable comments and suggestions.  ...

### Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing [chapter]

David R. Wood
2002 Lecture Notes in Computer Science
., 2001], who ask whether graphs of bounded tree-width have bounded queue-number?  ...  The minimum number of queues in a queue layout of a graph is its queue-number. Let be an Ò-vertex member of a proper minor-closed family of graphs (such as a planar graph).  ...  That is, graphs of bounded tree-width have bounded queue-number, and hence have three-dimensional drawings with linear volume.  ...

### Book Embeddings of Graph Products [article]

Sergey Pupyrev
2020 arXiv   pre-print
We show that the stack number is bounded for the strong product of a path and (i) a graph of bounded pathwidth or (ii) a bipartite graph of bounded treewidth and bounded degree.  ...  The stack number (book thickness, page number) of a graph is the minimum k such that it admits a k-stack layout.  ...   , yield a constant upper bound on the queue number of planar graphs.  ...

### The Local Queue Number of Graphs with Bounded Treewidth [article]

Laura Merker, Torsten Ueckerdt
2020 arXiv   pre-print
We present tools to bound the local queue number of graphs from above and below, focusing on graphs of treewidth k.  ...  Our results imply, inter alia, that the maximum local queue number among planar graphs is either 3 or 4.  ...  As the maximum average degree of planar graphs is strictly smaller than 6 and 2-trees are planar, Theorems 2 and 4 bound the maximum local queue number of the class of planar graphs. Corollary 6.  ...

### Layouts of Graph Subdivisions [chapter]

Vida Dujmović, David R. Wood
2005 Lecture Notes in Computer Science
This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.  ...  The stack-number (respectively, queue-number, track-number ) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.  ...  Thanks to Franz Brandenburg and Ulrik Brandes for pointing out the connection to double-ended queues.  ...

### Planar graphs have bounded queue-number [article]

Vida Dujmović, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, David R. Wood
2020 arXiv   pre-print
We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992.  ...  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.  ...  Analogues of Theorems 36 and 37 have been proved for bounded degree graphs in any minor-closed class  and for k-planar graphs and several other non-minor-closed classes of interest  .  ...

### Queue Layouts of Graphs with Bounded Degree and Bounded Genus [article]

Vida Dujmović and Pat Morin and David R. Wood
2019 arXiv   pre-print
Motivated by the question of whether planar graphs have bounded queue-number, we prove that planar graphs with maximum degree Δ have queue-number O(Δ^2), which improves upon the best previous bound of  ...  As a byproduct we prove that if planar graphs have bounded queue-number, then graphs of Euler genus g have queue-number O(g).  ...  We improve their bound and more generally show that graphs with bounded degree and bounded genus have bounded queue-number.  ...

### Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seaminar 19092)

Seok-Hee Hong, Michael Kaufmann, János Pach, Csaba D. Tóth, Michael Wagner
2019 Dagstuhl Reports
Next we discussed open research problems about beyond planar graphs, such as their combinatorial structures (e.g., book thickness, queue number), their topology (e.g., simultaneous embeddability, gap planarity  ...  This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory  ...  We thank Günter Rote and Martin Gronemann for asking interesting questions that led to some of this research.  ...

### Page 9208 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews
We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number.  ...  Math. 109 (2001), no. 3, 215-221; MR 2001m:05085], who asked whether graphs of bounded tree- width have bounded queue-number.  ...

### Characterisations and examples of graph classes with bounded expansion

Jaroslav Nešetřil, Patrice Ossona de Mendez, David R. Wood
2012 European journal of combinatorics (Print)
bounded queue number, and graphs with bounded non-repetitive chromatic number.  ...  In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with  ...  Acknowledgements The authors would like to thank Vida Dujmović for simplifying a clumsy proof in an early draft of the paper.  ...

### Planar Graphs have Bounded Queue-Number

Vida Dujmovic, Gwenael Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, David Wood
2019 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992.  ...  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.  ...  Acknowledgements This research was completed at the 7th Annual Workshop on Geometry and Graphs held at Bellairs Research Institute in March 2019.  ...

### Stacks, Queues and Tracks: Layouts of Graph Subdivisions

Vida Dujmović, David R. Wood
2005 Discrete Mathematics & Theoretical Computer Science
This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.  ...  \par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph.  ...  Thanks to Franz Brandenburg and Ulrik Brandes for pointing out the connection to double-ended queues. Thanks to Ferran Hurtado and Prosenjit Bose for graciously hosting the second author.  ...

### On Linear Layouts of Graphs

Vida Dujmović, David R. Wood
2004 Discrete Mathematics & Theoretical Computer Science
What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?  ...  \par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results.  ...  Planar graphs have arch-number at most three and this bound is tight. Proof. The Four Colour Theorem and Theorem 1 imply that all planar graphs have arch-number at most three.  ...

### Layouts of Expander Graphs [article]

Vida Dujmović, Anastasios Sidiropoulos, David R. Wood
2016 arXiv   pre-print
We then show that the same graphs admit 3-page book embeddings, 2-queue layouts, 4-track layouts, and have simple thickness 2. All these results are best possible.  ...  By combining this result with a generalisation of the unraveling method of Kannan, we construct 3-monotone bipartite expanders, which is best possible.  ...  Acknowledgement This research was initiated at the Workshop on Graphs and Geometry held at the Bellairs Research Institute in 2014. Thanks to the referee for the above observation.  ...

### Graph product structure for non-minor-closed classes [article]

Vida Dujmović and Pat Morin and David R. Wood
2022 arXiv   pre-print
It implies, amongst other results, that k-planar graphs have non-repetitive chromatic number upper-bounded by a function of k.  ...  ACM '20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path.  ...  bounds on the queue-number of map graphs.  ...
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