Thanks Jay and sorry I was not clear in my last comment.
I've applied your idea (method 2) on a network with 118 nodes and the last graph was a part of it. I followed your two steps. The red circle points are from step 1 and the red square points are from step 2.
Now, there are some facilities (the two red points) I can delete them while still I have for every node of the whole network at least two facilities.
For instance, delete node-39 and you will still have at least 2 facilities connected for each node;
Point 40 has 3 facilities connected to (40, 42 & 37),
Point 37 has 2 facilities (37, 40 & 34)
So, the final solution from the two steps is not a minimum optimal solution for locations of facilities because we can delete some of these facilities and get the objective of the problem.