Okay figured it out:

- First we check if the graph exists by checking if all the degrees of each vertex sum to an even number.

- We then follow RealiseNothing's approach.

- We take note that if the vertex with highest degree is equal to or greater than the amount of vertices's if this is true a simple graph does not exist.

- You eliminate the vertex with the higher degree and subtract 1 from each of the vertex degrees until the number of times you subtract equal to the degree of the vertex you deleted. (I don't think this needs an explanation)

- You keep doing this and if at any point the max degree is greater than or equal to the amount verticies then there is no simple graph.

- If you reach 0 then a simple graph exists

Example:

4, 4, 3, 2, 2, 1

3, 2, 1, 1, 1

1, 1

0

Therefore a simple graph exists.

5, 5, 3, 2, 2, 1

4, 2, 1, 1

Therefore since the highest degree is equal to the total number of vertices's there is no simple graph.

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