The [Erdős–Gallai theorem](http://en.wikipedia.org/wiki/Erd%25C5%2591s%25E2%2580%2593Gallai_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](http://www.jstor.org/discover/10.2307/2100346?uid=3739976&uid=2&uid=4&uid=3739256&sid=21101706626427) (1977).


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



Which degree sequences are planar graphical?
It is always difficult to say what is "currently known," but at least around
2008, the paper



> 
> "A Characterization of the degree sequences of 2-trees."
>  Prosenjit Bose, Vida Dujmovi, Danny Krizanc, Stefan Langerman, Pat Morin, David R. Wood, Stefanie Wuhrer.
>  *Journal of Graph Theory*
>  Volume 58, Issue 3, pages 191–209, July 2008.
>  ([journal link)](http://onlinelibrary.wiley.com/doi/10.1002/jgt.20302/abstract)
> 
> 
> 


gives a pretty definitive summary of the state of the art:



> 
> That a sequence of $n$ positive integers is the degree sequence of a tree if and only
>  if it sums to $2n − 2$ is a folklore result. Other graph
>  families with known degree sequence characterizations
>  include split graphs `[14, 20]`, $C\_4$-minor free graphs `[21]`,
>  unicyclic graphs `[1]`, cacti graphs `[15]`, Halin graphs `[2]`,
>  and edge-maximal outerplanar graphs `[18]`. The most
>  investigated class of graphs is that of planar graphs.
>  Despite the eﬀort, no characterization of the degree
>  sequences of planar graphs is known, even for edge-maximal planar graphs. 
>  Partial results are obtained in
>  `[5, 13, 11, 16]`.
> 
> 
> 


And, obviously, that paper characterizes [2-trees](http://en.wikipedia.org/wiki/K-tree).


