Constructing compact rectiliner planar layouts using canonical representation of planar graphs (original) (raw)

Optimal Polygonal Representation of Planar Graphs

Algorithmica, 2012

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1, -1. Fig. 1. Given a drawing of a planar graph(a), we apportion the edges to the endpoints by cutting each edge in half (b), and then apportion the faces to form polygons (c).

Floorplans, planar graphs, and layouts

IEEE Transactions on Circuits and Systems, 1988

The topics discussed in this paper are minimization of the area occupied by a layout and related results concerning networks flow and rectilinear representation of planar graphs, based on a graph model of floorplans and layouts. We do not restrict OUT analysis to sliced floorplans but allow arbitrary floorplans. Given an arbitrary floorplan and the areas of the embedded building blocks, we prove the existence and uniqueness of a zero wasted area layout, and characterize it by a necessary and sufficient condition. On the basis of this condition we develop a scheme to generate zero wasted area layouts. We prove that given a family of dual network pairs for which the product of dual arc lengths are invariant, the minimal product of their longest paths is not smaller than the maximal product of their shortest paths. We also show that the maximal product of the flows in such a family of dual network pairs is given by the total sum of the arc length product of each individual pair of dual arcs. Finally, based on the zero wasted area layout, we present an efficient procedure to derive a rectilinear representation for any planar graph.

Rectangular Drawings of Planar Graphs

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.

Box-Rectangular Drawings of Planar Graphs

Journal of Graph Algorithms and Applications, 2013

A plane graph is a planar graph with a fixed planar embedding in the plane. In a box-rectangular drawing of a plane graph, 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. A planar graph is said to have a box-rectangular drawing if at least one of its plane embeddings has a box-rectangular drawing. Rahman et al. [11] gave a necessary and sufficient condition for a plane graph to have a box-rectangular drawing and developed a lineartime algorithm to draw a box-rectangular drawing of a plane graph if it exists. Since a planar graph G may have an exponential number of planar embeddings, determining whether G has a box-rectangular drawing or not using the algorithm of Rahman et al. [11] for each planar embedding of G takes exponential time. Thus to develop an efficient algorithm to examine whether a planar graph has a box-rectangular drawing or not is a non-trivial problem. In this paper we give a linear-time algorithm to determine whether a planar graph G has a box-rectangular drawing or not, and to find a box-rectangular drawing of G if it exists.

Recognizing and Drawing IC-Planar Graphs

Lecture Notes in Computer Science, 2015

We give new results about the relationship between 1-planar graphs and RAC graphs. A graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is RAC if it can be drawn in such a way that its edges cross only at right angles. These two classes of graphs and their relationships have been widely investigated in the last years, due to their relevance in application domains where computing readable graph layouts is important to analyze or design relational data sets. We study ICplanar graphs, the sub-family of 1-planar graphs that admit 1-planar drawings with independent crossings (i.e., no two crossed edges share an endpoint). We prove that every IC-planar graph admits a straight-line RAC drawing, which may require however exponential area. If we do not require right angle crossings, we can draw every ICplanar graph with straight-line edges in linear time and quadratic area. We then study the problem of testing whether a graph is IC-planar. We prove that this problem is NPhard, even if a rotation system for the graph is fixed. On the positive side, we describe a polynomial-time algorithm that tests whether a triangulated plane graph augmented with a given set of edges that form a matching is IC-planar.

An algorithm for straight-line drawing of planar graphs

1998

Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs``nicely,''as it has the tendency to ruin the structure of the input graph.

On Rectilinear Drawing of Graphs

Lecture Notes in Computer Science, 2010

A rectilinear drawing is an orthogonal grid drawing without bends, possibly with edge crossings, without any overlapping between edges, between vertices, or between edges and vertices. Rectilinear drawings without edge crossings (planar rectilinear drawings) have been extensively investigated in graph drawing. Testing rectilinear planarity of a graph is NP-complete . Restricted cases of the planar rectilinear drawing problem, sometimes called the "no-bend orthogonal drawing problem", have been well studied (see, for example, ).

Efficient Enumeration of Drawings and Combinatorial Structures for Maximal Planar Graphs

arXiv (Cornell University), 2023

We propose efficient algorithms for enumerating the notorious combinatorial structures of maximal planar graphs, called canonical orderings and Schnyder woods, and the related classical graph drawings by de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical drawings and Schnyder drawings, respectively. To this aim (i) we devise an algorithm for enumerating special e-bipolar orientations of maximal planar graphs, called canonical orientations; (ii) we establish bijections between canonical orientations and canonical drawings, and between canonical orientations and Schnyder drawings; and (iii) we exploit the known correspondence between canonical orientations and canonical orderings, and the known bijection between canonical orientations and Schnyder woods. All our enumeration algorithms have O(n) setup time, space usage, and delay between any two consecutively listed outputs, for an n-vertex maximal planar graph.

2-visibility drawings of planar graphs

1997

In a 2-visibility drawing the vertices of a given graph are represented by rectangular boxes and the adjacency relations are expressed by horizontal and vertical lines drawn between the boxes. In this paper we want to emphasize this model as a practical alternative to other representations of graphs, and to demonstrate the quality of the produced drawings. We give several approaches, heuristics as well as provably good algorithms, to represent planar graphs within this model. To this, we present a polynomial time algorithm to compute a bend-minimum orthogonal drawing under the restriction that the number of bends at each edge is at most 1. 157

Complexity of Finding Non-Planar Rectilinear Drawings of Graphs

Lecture Notes in Computer Science, 2011

We study the complexity of the problem of finding nonplanar rectilinear drawings of graphs. This problem is known to be NPcomplete. We consider natural restrictions of this problem where constraints are placed on the possible orientations of edges. In particular, we show that if each edge has prescribed direction "left", "right", "down" or "up", the problem of finding a rectilinear drawing is polynomial, while finding such a drawing with the minimum area is NP-complete. When assigned directions are "horizontal" or "vertical" or a cyclic order of the edges at each vertex is specified, the problem is NP-complete. We show that these two NP-complete cases are fixed parameter tractable in the number of vertices of degree 3 or 4.