Hello,

1. This is how I would do the first one :

What you have to do is to try to reduce degrees gradually until you get a sequence of degrees which obviously are the degrees of a graph, for example (0,0,0,0,0). So, what we're actually doing is deleting the last degree , and then we subtract 1 from as many members as was the last number of sequence (the one that you 'deleted'). Also, you keep the sequence sorted all the time, like in the beginning. In your example it will be:

(1,2,2,2,3,3,4,4,4,5,5,7,7) ----> (1,2,2,2,2,3,3,3,3,4,4,6)

(1,2,2,2,2,3,3,3,3,4,4,6) ----> (1,2,2,2,2,2,2,2,2,3,3)

(1,2,2,2,2,2,2,2,2,3,3) ----> (1,1,1,2,2,2,2,2,2,2)

(1,1,1,2,2,2,2,2,2,2) ---> (1,1,1,1,1,2,2,2,2)

(1,1,1,1,1,2,2,2,2) ---> (1,1,1,1,1,1,1,2)

(1,1,1,1,1,1,1,2) ---> (0,0,1,1,1,1,1)

(0,0,1,1,1,1,1) ---> (0,0,0,1,1,1)

(0,0,0,1,1,1) ---> (0,0,0,0,1)

We got (0,0,0,0,1) which means that your sequence doesn't represent degrees of a graph. You can prove it in 2 ways: either showing that graph with 5 vertexes cannot have only one with degree 1, because every edge needs 2 vertex. Or you can just continue with this proces and show that in the next step you will get (-1,0,0,0), which is of course impossible, because vertex cannot have negative degree.