The [Erdős–Gallai theorem][1] characterizes which degree sequences are graphical (i.e. realizable by a simple graph).

There has been some work on which degree sequences are planar graphical (i.e. realizable by a simple planar graph).  See, for example, [On Planar Graphical Degree Sequences by Schmeichel and Hakimi][2] (1977).

What is currently known about which degree sequences are planar graphical?

  [1]: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Gallai_theorem
  [2]: https://www.jstor.org/stable/2100346 "doi:10.1137/0132048. zbMATH review at https://zbmath.org/0363.05029"