It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph.

[1][2][3] The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967.

When drawn on a plane, all its faces are triangular, making it a maximal planar graph.

As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph.

However, the Herschel graph, another non-Hamiltonian polyhedron with 11 vertices, has fewer edges.

[5] Based on the existence of such examples, Bernhart and Kainen conjectured that the book thickness of planar graphs could be made arbitrarily large, but it was subsequently shown that all planar graphs have book thickness at most four.

Geometrically, a polyhedron representing the Goldner–Harary graph may be formed by gluing a tetrahedron onto each face of a triangular dipyramid, similarly to the way a triakis octahedron is formed by gluing a tetrahedron onto each face of an octahedron.