TOPICS

# Self-Dual Graph

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 , , and , and tetrahedral graph .

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 appears to be the only regular self-dual graph.

Dual Graph, Self-Dual Polyhedron

## Explore with Wolfram|Alpha

More things to try:

## 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."

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