First off, let me say that you can find the answer to this question in Sage using the nauty generator. If you're going to be a serious graph theory student, Sage could be very helpful.
count = 0
for g in graphs.nauty_geng("20 180:180"):
count = count+1
print count
The answer is 4613. But, this isn't easy to see without a computer program.
At this point, perhaps it would be good to start by thinking in terms of of the number of connected graphs with at most 10 edges. Then, all the graphs you are looking for will be unions of these. You should be able to figure out these smaller cases. If any are too hard for you, these are more likely to be in some table somewhere, so you can look them up.
Connected graphs of order n and k edges is:
n = 1, k = 0: 1
n = 2, k = 1: 1
n = 3, k = 2: 1
n = 3, k = 3: 1
n = 4, k = 3: 2
n = 4, k = 4: 2
n = 4, k = 5: 1
n = 4, k = 6: 1
n = 5, k = 4: 3
n = 5, k = 5: 5
n = 5, k = 6: 5
n = 5, k = 7: 4
n = 5, k = 8: 2
n = 5, k = 9: 1
n = 5, k = 10: 1
.
.
.
n = 10, k = 9: 106
n = 10, k = 10: 657
n = 11, k = 10: 235
I used Sage for the last 3, I admit. But, I do know that the Atlas of Graphs contains all of these except for the last one, on P7.
One way you can approach this problem is noting that the vertices with degree 2 can be "reduced" to an edge without changing degrees. So we want to find all non-isomorphic connected simple graphs with degrees (3,3,3,3,4,4) first. Consider the complement of this graph; you get (1,1,2,2,2,2) as the degrees. There are only a few ways in which this can happen: you can have a lone edge and a 4-cycle, a 3-path and a 3-cycle, or a 6-path. (I believe these are the only cases)
Next, convert these back into graphs and the question is reduced to finding edges (which could be the same edge, giving a 3-long "edge") to "lengthen" by inserting a vertex in the middle.
Best Answer
Count the number of quadrilaterals and the number of pentagons in each graph. ( 4-cycles and 5-cycles )
You will come up with different numbers which indicate these are not isomorphic graphs.