Grid Drawings of 4-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 .

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⌋.

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.

Computational Geometry, 1998

Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in an (n-2) x (n-2) grid (for n ~> 3), and that no grid smaller than (2n/3-1) x (2n/3-1) can be used for this purpose, if n is a multiple of 3. In fact, for all n ~> 3, each dimension of the resulting grid needs to be at least [2(n-1)/3J, even if the other one is allowed to be unbounded. In this paper we show that this bound is fight by presenting a grid drawing algorithm that produces drawings of width 12(n-1)/33. The height of the produced drawings is bounded by 4/2(n-1)/3J-1. Our algorithm runs in linear time and is easy to implement.

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.

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.

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.

Computational Geometry: Theory and Applications, 2000

This paper addresses the problem of finding rectangular drawings of plane graphs, in which each vertex is drawn as a point, each edge is drawn as a horizontal or a vertical line segment, and the contour of each face is drawn as a rectangle. A graph is a 2–3 plane graph if it is a plane graph and each vertex has degree 3 except the vertices on the outer face which have degree 2 or 3. A necessary and sufficient condition for the existence of a rectangular drawing has been known only for the case where exactly four vertices of degree 2 on the outer face are designated as corners in a 2–3 plane graph G. In this paper we establish a necessary and sufficient condition for the existence of a rectangular drawing of G for the general case in which no vertices are designated as corners. We also give a linear-time algorithm to find a rectangular drawing of G if it exists.

Journal of Graph Algorithms and Applications, 2008

In a convex drawing of a plane graph, all edges are drawn as straightline segments without any edge-intersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n − 1) × (n − 1) grid if either G is triconnected or the triconnected component decomposition tree T (G) of G has two or three leaves, where n is the number of vertices in G. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n × n 2 grid if T (G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours.

Lecture Notes in Computer Science, 2002

A plane graph is a planar graph with a fixed embedding in the plane. In a rectangular drawing of a plane graph, each vertex is drawn as a point, each edge is drawn as a horizontal or vertical line segment, and each face is drawn as a rectangle. A planar graph is said to have a rectangular drawing if at least one of its plane embeddings has a rectangular drawing. In this paper we give a linear-time algorithm to examine whether a planar graph G of maximum degree three has a rectangular drawing or not, and to find a rectangular drawing of G if it exists.