Minimum-Area Drawings of

Journal of Graph Algorithms and Applications, 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 area of such a drawing is the 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 of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊ 2n 3 − 1⌋ × 2⌈ n 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3 ⌋×⌊ 2(n−1) 3 ⌋ for plane graphs.

2010

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 area of such a drawing is the 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 of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊2n n 3 − 1 ⌋ × 2⌈ 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1)

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.

Mathematics in Computer Science, 2011

A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment such that the line segments use only grid points as their endpoints. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G is a visibility drawing of G where the area is the minimum among all possible visibility drawings of G. The area minimization for grid visibility representation of planar graphs is NP-hard. However, the problem can be solved for a fixed planar embedding of a hierarchically planar graph in quadratic time. In this paper, we give a polynomial-time algorithm to obtain minimum-area visibility drawings of a plane 3-trees.

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.

Journal of Graph Algorithms and Applications, 2009

A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n 2). A lower bound of Ω(n 2) on the area-requirement for straight-line grid drawings of certain planar graphs are also known. In this paper, we introduce a fairly large class of planar graphs which admits a straight-line grid drawing on a grid of area O(n). We give a lineartime algorithm to find such a drawing. Our new class of planar graphs, which we call "doughnut graphs," is a subclass of 5-connected planar graphs. We show several interesting properties of "doughnut graphs" in this paper. One can easily observe that any spanning subgraph of a "doughnut graph" also admits a straight-line grid drawing with linear area. But the recognition of a spanning subgraph of a "doughnut graph" seems to be a non-trivial problem, since the recognition of a spanning subgraph of a given graph is an NP-complete problem in general. We establish a necessary and sufficient condition for a 4-connected planar graph G to be a spanning subgraph of a "doughnut graph." We also give a linear-time algorithm to augment a 4-connected planar graph G to a "doughnut graph" if G satisfies the necessary and sufficient condition.

Electronic Notes in Discrete Mathematics, 2008

We contribute to an open problem in Graph Drawing and improve the upper bound of the area of straight-line grid drawings of planar graphs to 4 3 n × 2 3 n. Our algorithm uses an improved version of the generic shift method [4] with one shift for each good vertex and two shifts for each bad vertex. The key is the handling of "critical vertices".

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⌋.

Journal of Combinatorial Optimization, 2013

A convex drawing of a plane graph G is a plane drawing of G, where each vertex is drawn as a point, each edge is drawn as a straight line segment and each face is drawn as a convex polygon. A maximal segment is a drawing of a maximal set of edges that form a straight line segment. A minimum-segment convex drawing of G is a convex drawing of G where the number of maximal segments is the minimum among all possible convex drawings of G. In this paper, we present a lineartime algorithm to obtain a minimum-segment convex drawing Γ of a 3-connected cubic plane graph G of n vertices, where the drawing is not a grid drawing. We also give a linear-time algorithm to obtain a convex grid drawing of G on an ( n 2 + 1) × ( n 2 + 1) grid with at most sn + 1 maximal segments, where sn = n 2 + 3 is the lower bound on the number of maximal segments in a convex drawing of G.