Classification and Construction of Planar, 3-Connected Kronecker Products

We give a complete classification of the Kronecker (i.e. direct) product graphs that are planar and $3$-connected (i.e. $3$-polytopal). They are all of the form \[H\wedge K_2,\] where $H$ is a $2$-connected graph, possibly non-planar, and satisfying specific properties that we will describe. Our proof is constructive, in the sense that we prescribe how to obtain all such graphs $H$, by adding a few edges in a specific way to a given planar, bipartite graph, that is either $3$-connected, or semi-hyper-$2$-connected. Moreover, for $H$ planar, we also give a more precise characterisation of this graph, regarding the number of its odd regions, and how they intersect. If $H\wedge K_2$ is a $3$-polytope, then we have $\delta(H\wedge K_2)=3$, so that the connectivity of $H\wedge K_2$ is $3$, and the connectivity of $H$ is either $2$ or $3$. We also briefly discuss which Cartesian and strong products are $3$-polytopal.