The.Algorithm.Design.Manual.Springer-Verlag.1998

Drawing Graphs Nicely
crossings or edge length, and thus significantly improve the drawing.
A good, general-purpose heuristic for drawing graphs models the graph as a system of springs and then
uses energy minimization to space the vertices. Let adjacent vertices attract each other with a force
proportional to the logarithm of their separation, while all nonadjacent vertices repel each other with a
force proportional to their separation distance. These weights provide incentive for all edges to be as
short as possible, while spreading the vertices apart. The behavior of such a system can be approximated
by determining the force acting on each vertex at a particular time and then moving each vertex a small
amount in the appropriate direction. After several such iterations, the system should stabilize on a
reasonable drawing. The input and output figures above demonstrate the effectiveness of the spring
embedding on a particular small graph.
If you need a polyline graph drawing algorithm, my recommendation is that you study several of the
implementations presented below, particularly graphEd and GraphViz, and see whether one of them can
do the job. You will have to do a significant amount of work before you can hope to develop a better
algorithm.
Once you have a graph drawn, this opens another can of worms, namely where to place the edge/vertex
labels. We seek to position the labels very close to the edges or vertices they identify, and yet to place
them such that they do not overlap each other or important graph features. Map labeling heuristics are
described in [WW95]. Optimizing label placement can be shown to be an NP-complete problem, but
heuristics related to bin packing (see Section ) can be effectively used.
Implementations: Georg Sander maintains a comprehensive WWW page on graph drawing at
http://www.cs.uni-sb.de/RW/users/sander/html/gstools.html. This is well worth checking out and
probably should be your first stop in hunting down programs for graph drawing.
The best ftp-able package of graph drawing algorithms is GraphEd, by Michael Himsolt. GraphEd
[Him94] is a powerful interactive editor that enables the user to construct and manipulate both directed
and undirected graphs. It contains a variety of graph and tree drawing algorithms, including planar
drawings, polyline drawings, upward drawings of directed acyclic graphs (DAGs), and spring
embeddings, and allows variations in node, edge, and label styles. Sgraph is an interface to GraphEd to
support user-specific extensions written in C. It includes a modest library of algorithms for planarity
testing, maximum flow, matching, and connectivity testing. GraphEd can be obtained by anonymous ftp
from forwiss.uni-passau.de (132.231.20.10) in directory /pub/local/graphed. GraphEd is free for
noncommercial use. Graphlet is a more recent project by the same group, available at http://www.fmi.unipassau.de/Graphlet.
GraphViz is a popular graph drawing program developed by Stephen North of Bell Laboratories. It
represents edges as splines and can construct useful drawings of quite large and complicated graphs. I
recommend it, even though licensing considerations make it impossible to include on the Algorithm
Repository or CD-ROM. A noncommercial license is available from http://portal.research.bell-
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