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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.  ...  [2] , yield a constant upper bound on the queue number of planar graphs.  ...

### 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 number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G.  ...  Thanks to Franz Brandenburg and Ulrik Brandes for pointing out the connection to double-ended queues.  ...

### 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.  ...

### 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.  ...  The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree.  ...  Acknowledgements The authors would like to thank Vida Dujmović for simplifying a clumsy proof in an early draft of the paper.  ...

### 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).  ...  It might even be true that graphs excluding a fixed minor have bounded queue-number.Planar graphs of bounded degree have constant queue number. 2018. STOC 2019, to appear.  ...

### 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.  ...  The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G.  ...  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.  ...

### 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 [46] and for k-planar graphs and several other non-minor-closed classes of interest [51] .  ...

### 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.  ...  This result is the first constant upper bound on the queue-number of bounded powers of graphs with bounded degree and bounded Euler genus.  ...

### Track Drawings of Graphs with Constant Queue Number [chapter]

Emilio Di Giacomo, Henk Meijer
2004 Lecture Notes in Computer Science
In [18] it is proved that every graph from a proper minor closed family has constant track number if and only if it has constant queue number.  ...  In this paper we study the track number of well-known families of graphs with small queue number.  ...  Acknowledgments We thank Giuseppe Liotta for useful discussions on the subject of this paper.  ...

### New Linear-Time Algorithms for Edge-Coloring Planar Graphs

Richard Cole, Łukasz Kowalik
2007 Algorithmica
We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max{Δ, 9} colors.  ...  Thus the coloring is optimal for graphs with maximum degree Δ ≥ 9. Moreover for Δ = 4, 5, 6 we give linear-time algorithms that use Δ + 2 colors.  ...  Acknowledgements The authors would like to thank anonymous referees for their excellent work and many useful suggestions which helped us avoid several mistakes and also improved the presentation of the  ...

### Finding Minimum Spanning Trees

David Cheriton, Robert Endre Tarjan
1976 SIAM journal on computing (Print)
A modification of the method improves the l+€ running time to 0(e), if e is fl(n ) for some positive constant 6. Another algorithm finds a minimum spanning tree of a planar graph in 0(n) time.  ...  This time bound is the same as that of a new algorithm by Yao, but Yao's method seems more complicated to implement.  ...  For all n, there is a graph of n vertices, with all vertices of degree three or more, having a girth which is fi(log n). Finding a planar representation for G requires 0(e) time [5] .  ...

### Layout of Graphs with Bounded Tree-Width

Vida Dujmovic, Pat Morin, David R. Wood
2005 SIAM journal on computing (Print)
This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings  ...  The minimum number of queues in a queue layout of a graph is its queue-number.  ...  Many other graph families have bounded acyclic chromatic number, including graphs embeddable on a fixed surface [2, 3, 6] , 1-planar graphs [13] , graphs with bounded maximum degree [5] , and graphs  ...
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