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.
