A self-dual graphs 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.

dual graph | self-dual polyhedron