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**11 - 14**of**14**### Queue Layouts and Three-Dimensional Straight-Line Grid Drawings

, 2002

"... A famous result due to de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and Schnyder [Order, 1989] states that every n-vertex planar graph has a (two-dimensional) straight-line grid drawing with O(n2) area. A three-dimensional straight-line grid drawing of a graph represents the vertices by gri ..."

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A famous result due to de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and Schnyder [Order, 1989] states that every n-vertex planar graph has a (two-dimensional) straight-line grid drawing with O(n2) area. A three-dimensional straight-line grid drawing of a graph represents the vertices by grid-points in 3-space and the edges by non-crossing line segments. This research is motivated by the following question of Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every planar graph have a three-dimensional straight-line grid drawing with O(n) volume? A queue layout con-sists of a linear order of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to . Let G be a mem-ber of a proper minor-closed family of graphs (such as a planar graph), and let F (n) be a set of functions closed under taking polynomials (such as O(1) or polylog n). We prove that G has a F (n) F (n) O(n) straight-line grid drawing if and only if G has a queue layout with F (n) queues. Thus the above question is closely related to the open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queuenumber? As a corollary we improve the best known upper bound on the volume of three-dimensional straight-line grid drawings of series-parallel graphs from O(n log2 n) to O(n).

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

, 2005

"... A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of ea ..."

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A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. 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. This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min{sn(G), qn(G)}). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number. 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. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. 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. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we

### Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions

"... This paper investigates the following question: Given a grid φ, where φ is a proper subset of the integer 2D or 3D grid, which graphs admit straight-line crossing-free drawings with vertices located at (integral) grid points of φ? We characterize the trees that can be drawn on a strip, i.e., on a tw ..."

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This paper investigates the following question: Given a grid φ, where φ is a proper subset of the integer 2D or 3D grid, which graphs admit straight-line crossing-free drawings with vertices located at (integral) grid points of φ? We characterize the trees that can be drawn on a strip, i.e., on a two-dimensional n × 2 grid. For arbitrary graphs we prove lower bounds for the height k of an n × k grid required for a drawing of the graph. Motivated by the results on the plane we investigate restrictions of the integer grid in 3D and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal – it supports all outerplanar graphs of n vertices. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to an n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.