Checking graph is simple or not ,given its degree sequence .

Question

I don't know what is the way to check this:

Check whether the graph having degree sequence $\{3,3,1,1\}$ is a simple graph or not?

Please help explaining the strategy I must follow to check this...

Answer 1

Let the vertices be $v_1,v_2,v_3$, and $v_4$, and suppose that $\deg v_1=\deg v_2=3$. If the graph is simple, and $\deg v_1=3$, $v_1$ must be adjacent to each of the other three vertices. The same is true of $v_2$. Thus, $v_3$ must be adjacent to $v_1$ and to $v_2$; can its degree be $1$? The argument is practically complete at this point, but I’ve included the rest in the spoiler-protected block below; mouse-over to see it.

Clearly it must have degree at least $2$. (The same is of course true for $v_4$.) Thus, $\langle 3,3,1,1\rangle$ is not the degree sequence of any simple graph.