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Graph Embedding


CubicalGraphEmbeddings

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.

CubicalGraphCircular

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.

GraphEmbeddings

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].

Hong and Eades (2003) gave a linear time algorithm for drawing disconnected planar graphs with maximum number of symmetries. Freivalds et al. (2002) gave an algorithm for embedding disconnected graphs based on polyomino packing.

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


See also

Circular Embedding, Embedding, Integral Embedding, Planar Embedding, Planar Straight Line Embedding, Rectilinear Crossing Number, Straight Line Embedding, Unit-Distance Graph

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References

Chung, F.; Leighton, T.; and Rosenberg, A. "Embeddings Graphs in Books: A Layout Problem with Applications to VLSI Design." SIAM J. Algebraic Disc. Meth. 8, 33-58, 1987.Di Battista, G.; Eades, P.; Tamassia, R.; and Tollis, I. G. Graph Drawing: Algorithms for the Visualization of Graphs. Englewood Cliffs, NJ: Prentice-Hall, 1998.Di Battista, G.; Garg, A.; Liotta, G.; Tamassia, R.; Tassinari, E.; and Vargiu, F. "An Experimental Comparison of Four Graph Drawing Algorithms." Computational Geom. 7, 303-325, 1997.Eades, P. "A Heuristic for Graph Drawing." Congr. Numer. 42, 149-160, 1984.Eades, P.; Fogg, I.; and Kelly, D. SPREMB: A System for Developing Graph Algorithms. Technical Report. Department of Computer Science. St. Lucia, Queensland, Australia: University of Queensland, 1988.Eades, P. and Tamassia, R. "Algorithms for Drawing Graphs: An Annotated Bibliography." Technical Report CS-89-09. Department of Computer Science. Providence, RI: Brown University, Feb. 1989.Freivalds, K.; Dogrusoz, U.; and Kikusts, P. "Disconnected Graph Layout and the Polyomino Packing Approach." In Graph Drawing: 9th International Symposium, GD 2001 Vienna, Austria, September 23-26, 2001, Revised Papers (Ed. P. Mutzel, M. Jünger, and S. Leipert). Berlin: Springer, pp. 378-391, 2002.Hong, S.-H. and Eades, P. "Symmetric Layout of Disconnected Graphs." In Algorithms and Computation: 14th International Symposium, ISAAC 2003, Kyoto, Japan, December 15-17, 2003, Proceedings (Ed. T. Ibaraki, N. Katoh, and H. Ono). Berlin: Springer, pp. 405-414, 2003.Kamada, T. and Kawai, S. "An Algorithm for Drawing General Undirected Graphs." Inform. Processing Lett. 31, 7-15, 1989.Malitz, S. M. "Genus g Graphs Have Pagenumber O(sqrt(g))." In Proc. 29th Sympos. Found. Computer Sci. IEEE Press, pp. 458-468, 1988.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.Vaucher, J. "Pretty Printing of Trees." Software Pract. Experience 10, 553-561, 1980.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.

Referenced on Wolfram|Alpha

Graph Embedding

Cite this as:

Weisstein, Eric W. "Graph Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GraphEmbedding.html

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