Is there any characterization on the set of integers $n$ such that there is a 3-connected 5-regular simple $n$-vertex planar graph?



There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....?
There is a 3-connected 5-regular simple $n$-vertex planar graph if and only if $n=12$ or $n \ge 16$ is even. See [Recursive generation of 5-regular graphs](https://link.springer.com/chapter/10.1007%2F978-3-642-00202-1_12) by Mahdieh Hasheminezhad, Brendan D. McKay, Tristan Reeves in *WALCOM: Algorithms and Computation*, eds. Das and Uehara, *Lecture Notes in Computer Science*, vol 5431, Springer 2009. The number of such graphs is given in [OEIS A308489](https://oeis.org/A308489).


They use a set of 7 graphs that are irreducible under a system of expansions & reductions and, as is common for contemporary graph theory, computer assistance. E.g., "The program completed execution in 21 seconds. In total, 39621 induced subgraphs were found..."


