Computing straight-line 3D grid drawings of graphs in linear volume
Computational Geometry: Theory and Applications
Queue layouts of iterated line directed graphs
Discrete Applied Mathematics
An improved upper bound on the queuenumber of the hypercube
Information Processing Letters
(3,2)-Track Layout of Bipartite Graph Subdivisions
Computational Geometry and Graph Theory
(d+1,2) -Track Layout of Bipartite Graph Subdivisions
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
European Journal of Combinatorics
Computing straight-line 3D grid drawings of graphs in linear volume
Computational Geometry: Theory and Applications
Upper bounds on the queuenumber of k-ary n-cubes
Information Processing Letters
Drawing Kn in three dimensions with one bend per edge
GD'05 Proceedings of the 13th international conference on Graph Drawing
Volume requirements of 3d upward drawings
GD'05 Proceedings of the 13th international conference on Graph Drawing
Characterisations and examples of graph classes with bounded expansion
European Journal of Combinatorics
GD'04 Proceedings of the 12th international conference on Graph Drawing
GD'04 Proceedings of the 12th international conference on Graph Drawing
Small drawings of series-parallel graphs and other subclasses of planar graphs
GD'09 Proceedings of the 17th international conference on Graph Drawing
Acyclic colorings of graph subdivisions
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Acyclically 3-colorable planar graphs
Journal of Combinatorial Optimization
SIAM Journal on Discrete Mathematics
Acyclic colorings of graph subdivisions revisited
Journal of Discrete Algorithms
h-quasi planar drawings of bounded treewidth graphs in linear area
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Acyclic coloring with few division vertices
Journal of Discrete Algorithms
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by noncrossing line-segments. This paper contributes three main results:(1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a $\mathcal{O}(1) \times \mathcal{O}(1) \times \mathcal{O}(n)$ drawing if and only if G has a $\mathcal{O}(1)$ queue-number.(2) It is proved that the queue-number is bounded by the tree-width, thus resolving an open problem due to Ganley and Heath [Discrete Appl. Math., 109 (2001), pp. 215--221] and disproving a conjecture of Pemmaraju [Exploring the Powers of Stacks and Queues via Graph Layouts, Ph. D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1992]. 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 with $\mathcal{O}(n)$ volume. This is the most general family of graphs known to admit three-dimensional drawings with $\mathcal{O}(n)$ volume.The proofs depend upon our results regarding track layouts and tree-partitions of graphs, which may be of independent interest.