Graph Drawing Problem Research Articles

Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems

In this paper we extend the concept of the regular edge labeling for general plane graphs and for triconnected triangulated plane graphs to 4-connected triangulated plane graphs. We present two different linear time algorithms for constructing such a labeling. By using regular edge labeling, we present a new linear time algorithm for constructing rectangular dual of planar graphs. Our algorithm is simpler than previously known algorithms. The coordinates of the rectangular dual constructed by our algorithm are integers, while the one constructed by known algorithms are real numbers. Our second regular edge labeling algorithm is based on canonical ordering of 4-connected triangulated plane graphs. By using this technique, we present a new algorithm for constructing visibility representation of 4-connected planar graphs. Our algorithm reduces the size of the representation by a factor of 2 for such graphs.

Drawing planar graphs using the canonical ordering

We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows: Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n−4)×(n−2) grid, wheren is the number of vertices. Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends. Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid. Every triconnected planar graphG admits a planar polyline grid drawing on a (2n−6)×(3n−9) grid with minimum angle larger than 2/d radians and at most 5n−15 bends, withd the maximum degree.

Drawing Planar Graphs Using the Canonical Ordering

We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows: Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n−4)×(n−2) grid, wheren is the number of vertices. Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends. Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid. Every triconnected planar graphG admits a planar polyline grid drawing on a (2n−6)×(3n−9) grid with minimum angle larger than 2/d radians and at most 5n−15 bends, withd the maximum degree. These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g., concerning visibility representations, are included.

A programming project

Graph drawing problems provide excellent material for programming projects. As an example, this paper describes the results of an undergraduate project which dealt with hypergraph drawing. We introduce a practical method for drawing hypergraphs. The method is based on the spring algorithm, a well-known method for drawing normal graphs.

An automatic layout facility (abstract)

We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. With this method linear time an space algorithms can be designed for many graph drawing problems. (i) We show that every triconnected planar graph G can be drawn convexly with straight lines on an (2 n -4)×( n -2) grid. (ii) If G has maximum degree four (three), then G can be drawn orthogonal with at most [3 n /2] + 3 (at most [ n /2] + 1) bends on an n × n grid ([ n /2] × [ n /2] grid, respectively). (iii) If G has maximum degree d , then G can be drawn planar on an (2 n -6) × (3 n -6) grid with minimum angle larger than 1/d-2 radians and at most 5 n -15 bends. These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g. concerning visibility representations, are included.

Fixed edge-length graph drawing is NP-hard

The problem of drawing a graph with prescribed edge lengths such that edges do not cross is proved NP-hard, even in the case where all edge lengths are one and the graph is 2-connected. The proof is an interesting interplay of geometry and combinatorics. First we use geometrical methods to transform a rather synthetic “Orientation Problem” to our graph drawing problem; then we use combinatorial methods to transform a well-known “Flow Problem” to the orientation problem.

Graph graphics: Theory and practice

Many graph drawing problems are NP-complete. Most of the problems described in this expository survey have constraints that relate to either minimizing the number of crossings or minimizing some function of the edge lengths. Nevertheless, these problems are sufficiently common in some areas that polynomial-time algorithms for special cases and heuristics for more general cases have been developed. A graph-theoretic characterization of some of these problems is given, followed by a discussion of their complexity. Finally, several algorithms and heuristics for graph layout [some with applications to very large scale integration (VLSI)] will be discussed.

Genetic Graph Drawing

The drawing of graphs is widely recognized as a very important task in diverse elds of research and development. Examples include VLSI Design and plant layout. Also, many CASE methodologies and interactive dataintensive applications should perform graph drawing and should manage a large amount of constraints including semantic ones. On the other hand, when the information that must be visualized is changing constantly, is necessary to develop automatic methods to partially (or totally) redraw the diagrams. We want to obtain drawings that are optimal in the sense of readability. However, is di cult to create a general-purpose graph drawer and is also computationally expensive. Genetic algorithms (GAs) have recently received a growing interest and have been applied in many design problems. In this paper we describe why the GAs-based approach is a good alternative to obtain a general exible tool for Graph Drawing Problems, dealing also with semantic aesthetics. Free Form Deformation (FFD) has been used in structural design problems in combination with GAs. We also present an approach to automatic graph drawing based on GAs and FFD, for improving of a given layout by exploring its neighborhood. The search space is reduced and the readability of the drawings is improved. Also the Mental Map is almost preserved. Transactions on Information and Communications Technologies vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-3517