GraphDraw—A Tool for the Represention of Graphs Using Inherent Symmetry ^†

When drawing small graphs (with up to five or six vertices) we can use their underlying symmetry to represent them in a clear and natural way. For example, to draw a graph G with vertex set V = {0,1,2,3} and edge set E = {(0,1), (1,2), (2,3), (3,0)} it would be natural to use a square, or if V = {0,1,2,3,4} and E = {(0,2), (0,3),(1,3),(1,4),(2,4)} we might use a star to represent G. Even if the vertex labels were permuted, it would not be difficult to uncover the underlying shape. As the number of vertices increases, it becomes impossible to manually choose a suitable and informative representation, i.e., to choose where on the plane to place each of the vertices in order to produce a clear graph, rather than just a confusing mess. Existing graph drawing applications (like yEd) require the user to decide where to place the vertices.

In this paper, we present a tool—GraphDraw—which uses the underlying automorphism group of a graph (A(G)) to draw graphs from an initial index array representation of G. Cycles in the generators of A(G) are used to place the vertices in such a way as to exploit the underlying symmetry in the drawing, thus producing a clearer and more intuitive representation. Parameter selection allows the user to choose from a range of representations and to optimise the graphs in terms of edge crossings or total edge length. We give a description of the tool and present a suite of example graphs, illustrating the effect of different parameter selections.