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Self-Dual Graph

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SelfDualGraphs

A self-dual graph is a graph that is dual to itself. Wheel graphs are self-dual, as are the examples illustrated above. Naturally, the skeleton of a self-dual polyhedron is a self-dual graph. Since the skeleton of a pyramid is a wheel graph, it follows that pyramids are also self-dual.

Additional self-dual graphs include the Goddard-Henning graph, skeletons of the Johnson solids J_7, J_8, and J_9, and tetrahedral graph K_4=W_4.

The numbers of self-dual polyhedral graphs on 1, 2, ... vertices are 0, 0, 1, 1, 2, 6, 16, 50, 165, 554, 1908, ... (OEIS A002841).

The tetrahedral graph K_4 appears to be the only regular self-dual graph.

See also

Dual Graph, Self-Dual Polyhedron

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References

House of Graphs. "Planar Graphs: 3-Connected Planar Self-Dual Graphs." https://hog.grinvin.org/Planar#selfdual.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976.Smith, C. A. B. and Tutte, W. T. "A Class of Self-Dual Maps." Canad. J. Math. 2, 179-196, 1950.Sloane, N. J. A. Sequence A0028411615 in "The On-Line Encyclopedia of Integer Sequences."

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Self-Dual Graph

Cite this as:

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

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