On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces

Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the projective plane, the torus and the Klein bottle each are in one-to-one correspondence with certain subgraphs of the dual graph $G^\ast$. Here, we exploit this characterisation and show that two strong embeddings of $G$ on the projective plane, the torus or the Klein bottle are isomorphic if and only if the corresponding subgraphs of $G^{\ast}$ are contained in the same orbit under $\mathrm{Aut}(G^{\ast})$. This allows us to construct a data base containing all isomorphism classes of strong embeddings on the projective plane, the torus and the Klein bottle of all 3-connected cubic planar graphs with up to 22 vertices. Moreover, we establish that cyclically 4-edge connected cubic planar graphs can be strongly embedded on orientable surfaces of positive genera. We use this to show that a 3-connected cubic planar graph has no strong embedding on orientable surfaces of positive genera if and only if it is the dual of an Apollonian network.