An algorithm for straight-line drawing of planar graphs

sharif.ac.ir

Layered graph drawing [1, 14] is a popular paradigm for drawing graphs which has applications in visual-ization [15], in DNA mapping, and in VLSI layout [9]. In a layered drawing of a graph, vertices are arranged in horizontal layers, and edges are routed as polygonal lines ...

Mathematics and Visualization, 2004

Graph Algorithms and Applications 4, 2006

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.

2011

The classical Fáry's theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry's theorem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1planar graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. We also show that there are 1-planar graphs for which every straight-line drawing has exponential area. To our best knowledge, this is the first result to extend Fáry's theorem to non-planar graphs.

Lecture Notes in Computer Science, 2014

In this paper we study the ANCHORED GRAPH DRAWING (AGD) problem: Given a planar graph G, an initial placement for its vertices, and a distance d, produce a planar straight-line drawing of G such that each vertex is at distance at most d from its original position. We show that the AGD problem is NP-hard in several settings and provide a polynomial-time algorithm when d is the uniform distance L∞ and edges are required to be drawn as horizontal or vertical segments.

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.

Journal of Graph Algorithms and Applications, 2013

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straight-line drawings of G. In this paper we prove that it is NP-complete to determine whether a plane graph G has a straight-line drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straight-line drawing with at most k segments is NP-complete, even when G is an outerplanar graph. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.

Proceedings of the eleventh annual symposium on Computational geometry - SCG '95, 1995

In this paper we present an extensive experimental study comparing three general-purpose graph drawing algorithms. The three algorithms take as input general graphs (with no restrictions whatsoever on the connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in \real-life" software engineering and database applications. The experiments provide a detailed quantitative evaluation of the performance of the three algorithms, and show that they exhibit trade-o s between \aesthetic" properties (e.g., crossings, bends, edge length) and running time. The observed practical behavior of the algorithms is consistent with their theoretical properties.

2012

Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non-commercial use, distribution, and reproduction inany medium, provided the original work is properly cited. Global Journal of Computer Science and Technology Graphics & Vision Volume 12 Issue 13 Version 1.0 Year 2012 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 0975-4172 & Print ISSN: 0975-4350

IOSR Journal of Computer Engineering, 2013

This paper presented study on convex drawing of planar graph. In graph theory, a planar graph is a graph that can be embedded in the plane. A planar graph is one that can be drawn on a plane in such a way that there are no "edge crossings," i.e. edges intersect only at their common vertices. Convex polygon has all interior angles less than or equal to 180°. A graph is called a convex drawing if every facial cycle (face) is drawn as a convex polygon. In a convex drawing of a planar graph, all edges are drawn by straight line segments in such a way that every face boundary in a convex polygon. This paper describes some of the recent works on convex drawing on planar graph.

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

Lecture Notes in Computer Science, 2013

Given a plane graph G (i.e., a planar graph with a fixed planar embedding) and a simple cycle C in G whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of G. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time.

Discrete & Computational Geometry, 1986

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graph G = (V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs in O(I VI 2) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-andconquer algorithm based on the polygon cutting theorem of Chazelle is given that runs in O(I V I logl VI) time.

1995

In this paper we present an extensive experimental study comparing four general-purpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in "real-life" software engineering and database applications. The experiments