We describe a four-pass algorithm for drawing directed graphs. The first pass finds an optimal rank assignment using a network simplex algorithm. The second pass sets the vertex order within ranks by an iterative heuristic incorporating a novel weight function and local transpositions to reduce crossings. The third pass finds optimal coordinates for nodes by constructing and ranking an auxiliary graph. The fourth pass makes splines to draw edges. The algorithm makes good drawings and runs fast. 1.

ABSTRACT — In this paper, we develop a greedy randomized adaptive search procedure (GRASP) for the problem of minimizing straight-line crossings in a 2-layer graph. The procedure is fast and is particularly appealing when dealing with low-density graphs. When a modest increase in computational time is allowed, the procedure may be coupled with a path relinking strategy to search for improved outcomes. Although the principles of path relinking have appeared in the tabu search literature, this search strategy has not been fully implemented and tested. We perform extensive computational experiments with more than 3,000 graph instances to first study the effect of changes in critical search parameters and then to compare the efficiency of alternative solution procedures. Our results indicate that graph density is a major influential factor on the performance of a solution procedure. Laguna and Martí / 2 The problem of minimizing straight-line crossings in layered graphs has been the subject of study for at least 17 years, beginning with the Relative Degree Algorithm introduced by Carpano [2]. The problem consists of aligning the two shores V1 and V2 of a bipartite graph G = (V1, V2, E) on two parallel straight lines (layers) such that the number of crossing between the edges in E is minimized

The VCG tool allows the visualization of graphs that occur typically as data structures in programs. We describe the functionality of the VCG tool, its layout algorithm and its heuristics. Our main emphasis in the selection of methods is to achieve a very good performance for the layout of large graphs. The tool supports the partitioning of edges and nodes into edge classes and nested subgraphs, the folding of regions, and the management of priorities of edges. The algorithm produces good drawings and runs reasonably fast even on very large graphs.

dag is a pic or POSTSCRIPT preprocessor that draws directed graphs. It works well on acyclic graphs and other graphs that can be drawn as hierarchies. Graph descriptions contain nodes, edges, and optional control statements. Here is a drawing of a graph from Forrester's book, World Dynamics (Wright-Allen, Cambridge, MA, 1971). It took 2.1 CPU seconds on a VAX-8650 to make this drawing. S8 9 S24 25 27 S1 2 10 S35 43 36 S30 31 33 42 T1 26 T24 3 16 17 18 11 14 13 12 32 T30 34 4 15 19 29 37 39 41 38 40 23 5 21 20 28 6 T35 22 7 T8 I. Introduction Directed graphs have many applications in computing, such as describing data structures, finite automata, data flow, procedure calls, and software configuration dependencies. A picture is a good way to represent a directed graph. It is seldom easy to understand much about a graph from a list of edges, but with a picture one can quickly find individual nodes, groups of related nodes, and trace paths in the graph. The main obstacle is that it can be...

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

Several applications use algorithms for drawing k-layered networks and, in particular, 2-layered networks (i.e. bipartite graphs). Bipartite graphs are commonly drawn in the plane so that all vertices lie on two parallel vertical lines, and an important requirement in drawing such graphs is to minimize edge crossings. Such a problem is NP-complete even when the position of the vertices on one layer is held fixed. This paper presents a heuristic, called the assignment heuristic, for edge crossing minimization in bipartite graphs, which works by reducing the problem to an assignment problem. The main idea of the assignment heuristic is to position simultaneously all the vertices of one layer, so that the mutual interaction of the position of all the vertices can be taken into account. We also show that the idea underlying the assignment heuristic can be effectively applied in other cases requiring edge crossing minimization. Index Terms - Graph layout, crossing reduction, bipartite gra...

This paper introduces genetic algorithms for the level permutation problem (LPP). The problem is to minimize the number of edge crossings in a bipartite graph when the order of vertices in one of the two vertex subsets is fixed. We show that genetic algorithms outperform the previously known heuristics especially when applied to low density graphs. Values for various parameters of genetic LPP algorithms are tested.

We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1-SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.

Abstract — Graphs are commonly used as a basic modeling tool in areas such as project management, production scheduling, line balancing, business process reengineering, and software visualization. An important problem in the area of graph drawing is to minimize arc crossings in a multi-layer hierarchical digraph. Existing solution methods for this problem are based on simple ordering rules for single layers that may lead to inferior drawings. This paper first introduces an extensive review of relevant work previously published in this area. Then a tabu search implementation is presented that seeks high-quality drawings by means of an intensification phase that finds a local optimum according to an insertion mechanism and two levels of diversification. Computational experiments with 200 graphs with up to 30 nodes per layer and up to 30 layers are presented to assess the merit of the method.

Graph drawing algorithms usually attempt to display the characteristic properties of the input graphs. In this paper we consider the class of planar bipartite graphs and try to achieve planar drawings such that the bipartiteness property is cleary shown. To this aim, we develop several models, give efficient algorithms to find a corresponding drawing if possible or prove the hardness of the problem. 1 Introduction Graph drawing is a more and more developing method to visualize data and their relations. The main goal is to draw the graph in such a way that certain properties are clearly displayed: Planar graphs should be drawn planar [6], symmetries should be displayed [3, 11], if the graph is directed and acyclic then it should be drawn 'upward' [12], cliques should be easily recognized. There are many more properties developed in graph theory and graph algorithms that are worth to be displayed [1]. Another important property is the bipartiteness. A graph G = (V; E) with a set V of v...