Minimum-width grid drawings of plane graphs (original) (raw)
Related papers
Rectangular grid drawings of plane graphs
Computational Geometry: Theory and Applications, 1996
The rectangular grid drawing of a plane graph G is a drawing of G such that each vertex is located on a grid point, each edge is drawn as a horizontal or vertical line segment, and the contour of each face is drawn as a rectangle. In this paper we give a simple linear-time algorithm to find a rectangular grid drawing of G if it exists. We also give an upper bound on the sum of required width W and height H and a bound on the area of a rectangular grid drawing of G, where n is the number of vertices in G. These bounds are best possible, and hold for any compact rectangular grid drawing.
Grid Drawings of Four-Connected Plane Graphs
Lecture Notes in Computer Science, 1999
A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm takes time O(n) and needs a rectangular grid of width n/2 − 1 and height n/2 if G has n vertices. The algorithm is best possible in the sense that there are an infinite number of 4-connected plane graphs any grid drawings of which need rectangular grids of width n/2 − 1 and height n/2 .
Grid Drawings of 4-Connected Plane Graphs
Discrete & Computational Geometry, 2001
A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm takes time O(n) and yields a drawing in a rectangular grid of width n/2 − 1 and height n/2 if G has n vertices. The algorithm is best possible in the sense that there are an infinite number of 4-connected plane graphs, any grid drawings of which need rectangular grids of width n/2 − 1 and height n/2 .
Convex Grid Drawings of Four-Connected Plane Graphs
International Journal of Foundations of Computer Science, 2006
A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.
2011
A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The height, width and area of such a drawing are respectively the height, width and area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area is the minimum. It is NP-complete to determine whether a plane graph G has a straight-line grid drawing with a given area or not. In this paper we give a polynomial-time algorithm for finding a minimum-area drawing of a plane 3-tree. Furthermore, we show a ⌊ 2n 3 −1⌋×2⌈n ⌉ lower bound for the area of a straight-line grid drawing of 3 a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3
Convex Grid Drawings of 3-Connected Planar Graphs
International Journal of Computational Geometry & Applications, 1997
We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.
Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions
Journal of Graph Algorithms and Applications, 2003
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. Communicated by: P. Mutzel and M. Jünger;
Monotone Grid Drawings of Planar Graphs
Lecture Notes in Computer Science, 2014
A monotone drawing of a planar graph G is a planar straightline drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of planar graphs have been proposed as a new standard for visualizing graphs. A monotone drawing of a planar graph is a monotone grid drawing if every vertex in the drawing is drawn on a grid point. In this paper we study monotone grid drawings of planar graphs in a variable embedding setting. We show that every connected planar graph of n vertices has a monotone grid drawing on a grid of size O(n) × O(n 2), and such a drawing can be found in O(n) time.
Straight-Line Grid Drawings of 3-Connected 1-Planar Graphs
Lecture Notes in Computer Science, 2013
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. In general, 1-planar graphs do not admit straightline drawings. We show that every 3-connected 1-planar graph has a straight-line drawing on an integer grid of quadratic size, with the exception of a single edge on the outer face that has one bend. The drawing can be computed in linear time from any given 1-planar embedding of the graph.
Box-Rectangular Drawings of Plane Graphs
Journal of Algorithms, 2000
In this paper we introduce a new drawing style of a plane graph G, called a "box-rectangular drawing." It is defined to be a drawing of G on an integer grid such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. We establish a necessary and sufficient condition for the existence of a box-rectangular drawing of G. We also give a simple lineartime algorithm to find a box-rectangular drawing of G if it exists.