- Sort the degrees in non-increasing order.
- Pick up the highest degree ( let us say it a), remove it from the list of degrees and subtract $1$ from next a degrees in list.
- Repeat Step $2$ until :
- If we get all $0$ entries in list $\color{red}{\Rightarrow}$ Simple Graph exits
- If we get a negative entry or not enough entries to subtract $1$ in step 2 $\color{red}{\Rightarrow}$ Simple Graph does not exist
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Let's take a example : $3, 2, 1, 2$
Step 1 : Sort the degree sequence : $3, 2, 2, 1$
Step 2: Pick $3$, Remove $3$ from list and from next $3$ elements subtract $1$, $ Result : (1,1,0) $
Again repeat step 2 : select $1$, Remove $1$ and from next $1$ subtract $1$, $Result : (0,0,0) $
Thus, a simple graph exists for the following degree-sequence.