TITLE: List of ways to tell if degree sequence is impossible for a simple graph
QUESTION [12 upvotes]: I'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$  is not possible because there are only 3 vertices in total so one can't have degree 3.
The list so far

vertices has degree equal to or larger than number of vertices
sum of degrees is odd
for n vertices if one has degree n-1 and another has degree 0
for n vertices the sum of the degrees cannot be greater than $n(n-1)$ because this would be have more edges than a complete graph

REPLY [6 votes]: Three is Havel–Hakimi algorithm  
It is stated by Wikipedia as:

The algorithm
  The algorithm is based on the following theorem.
  Let $$S=(d_{1},\dots ,d_{n})$$ be a finite list of nonnegative integers that is nonincreasing. List S is graphic if and only if the finite list $$S'=(d_{2}-1,d_{3}-1,\dots ,d_((d_{1}+1))-1,d_((d_{1}+2)),\dots ,d_{n})$$ has nonnegative integers and is graphic.
  If the given list S graphic then the theorem will be applied at most $n-1$ times setting in each further step $$S:=S'$$. Note that it can be necessary to sort this list again. This process ends when the whole list S' consists of zeros. In each step of the algorithm one constructs the edges of a graph with vertices $$v_{1},\cdots ,v_{n}$$, i.e. if it is possible to reduce the list $S$ to $S'$, then we add edges $$\{v_{1},v_{2}\},\{v_{1},v_{3}\},\cdots ,\{v_{1},v_((d_{1}+1))\}$$. When the list S cannot be reduced to a list S' of nonnegative integers in any step of this approach, the theorem proves that the list S from the beginning is not graphic.   

see also
What condition need to be imposed on Havel-Hakimi theorem to check for connected graph?