This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions 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. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. A three-dimensional (straight-line) grid-drawing of a graph represents the vertices by distinct points in Z 3, and represents each edge by a line-segment between its endpoints that does not intersect any other vertex, and does not intersect any other edge except at the endpoints. A folklore result states that every (simple) graph has a three-dimensional grid-drawing (see [2]). We therefore are interested in grid-drawings with small ‘volume’. The bounding box of a three-dimensional grid-drawing is the axis-aligned box of minimum size that contains the drawing. By an X × Y × Z grid-drawing we mean a three-dimensional griddrawing, such that the edges of the bounding box contain X, Y, and Z grid-points, respectively. The volume of a three-dimensional grid-drawing is the number of grid-points in the bounding box; that is, the volume of an X ×Y ×Z grid-drawing is XY Z. (This definition is formulated to ensure that a two-dimensional grid-drawing has positive volume.) Our main contribution is the following extremal result. Theorem 1. The maximum number of edges in an X × Y × Z grid-drawing is exactly (2X − 1)(2Y − 1)(2Z − 1) − XY Z. Proof. Consider an X × Y × Z grid-drawing of a graph G with n vertices and m edges. Let P be the set of points (x, y, z) contained in the bounding box such that 2x, 2y, and 2z are all integers. Observe that |P | = (2X − 1)(2Y − 1)(2Z − 1). The midpoint of every edge of G is in P, and no two edges share a common midpoint. Hence m ≤ |P |. In addition, the midpoint of an edge does not intersect a vertex. Thus m ≤ |P | − n. (1) A drawing with the maximum number of edges has no edge that passes through a grid-point. Otherwise, sub-divide the edge, and place the new vertex at that grid-point. Thus n = XY Z, and m ≤ |P | − XY Z, as claimed. This bound is attained by the following construction. Associate a vertex with each grid-point in an X × Y × Z grid-box B. As illustrated in Figure 1, every vertex (x, y, z) is adjacent to each

A box is a restricted portion of the three-dimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A box-drawing of a graph is a straight-line crossing-free drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with tri-connected components do not admit a box-drawing. This paper shows that even structurally simpler planar graphs, namely series-parallel graphs, are not boxdrawable in general. On the positive side, it is proved that every series-parallel graph whose vertices have maximum degree at most three is box-drawable. A drawing algorithm is presented that computes a box drawing of a 3-planar series-parallel graph in optimal time and with optimal volume.

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was O(n ). These results (partially) solve open problems due to Pach, Thiele, and Toth (1997) and Felsner, Liotta, and Wismath (2001).

Abstract. A tree-partition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every k-tree has a tree-partition such that each bag induces a (k − 1)-tree, amongst other properties. Applications of this result to two well-studied models of graph layout are presented. First it is proved that graphs of bounded tree-width have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded tree-width have three-dimensional straight-line grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1

Abstract. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z 3 and the edges by non-crossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order σ 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 σ. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree. 1

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We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

www.elsevier.com/locate/comgeo It is proved that every k-colourable graph on n vertices has a grid drawing with O(kn) area, and that this bound is best possible. This result can be viewed as a generalisation of the no-three-in-line problem. A further area bound is established that includes the aspect ratio as a parameter. © 2004 Elsevier B.V. All rights reserved.