Graph Embedding

A graph embedding, sometimes also called a graph drawing, is a particular drawing of a graph. Graph embeddings are most commonly drawn in the plane, but may also be constructed in three or more dimensions. The above figure shows several embeddings of the cubical graph. The most commonly encountered graph embeddings are generally straight line embeddings, in which all edges are drawn as straight line segments.

A good choice of embedding can lead to particularly illuminating diagrams. For example, the circular (left) embedding of the cubical graph illustrates this graph's inherent symmetries.

Skiena (1990) considers a number of different types of embeddings, including circular, ranked, radial, rooted, and spring. Graph embeddings can be visualized in the Wolfram Language in two dimensions using the option GraphLayout. Alternately, GraphPlot[g] can be used in two dimensions and GraphPlot3D[g] in three dimensions. Embeddings for trees can be visualized using TreePlot[g].

Precomputed embeddings of certain types for a number of graphs are available in the Wolfram Language as GraphData[g, "Graph", type].

REFERENCES

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Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Cambridge, England: Cambridge University Press, 2003.Reingold, E. and Tilford, J. "Tidier Drawings of Trees." IEEE Trans. Software Engin. 7, 223-228, 1981.

Skiena, S. "Graph Embeddings." §3.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 81 and 98-118, 1990.

Supowit, K. and Reingold, E. "The Complexity of Drawing Trees Nicely." Acta. Inform. 18, 377-392, 1983.

Tamassia, R. "Graph Drawing." Ch. 21 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 937-971, 2000.

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Wetherell, C. and Shannon, A. "Tidy Drawings of Trees." IEEE Trans. Software Engin. 5, 514-520, 1979.

White, A. T. "Imbedding Problems in Graph Theory." Ch. 6 in Graphs of Groups on Surfaces: Interactions and Models (Ed. A. T. White). Amsterdam, Netherlands: Elsevier, pp. 49-72, 2001.