The 12-vertex graph consisting of a cube in which two opposite faces (say, top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are perpendicular to each other.
The bidiakis cube is implemented in the Wolfram Language as GraphData["CubicPolyhedral", 4].
The bidiakis cube is biconnected, cubic, polyhedral, and Hamiltonian. It has three LCF notations: , , and .