The Erdős–Gallai theorem 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 (1977).

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