Choose $n/2$ vertices and guarantee $3/4$ of edges are accounted for proof
I'm not sure your algorithm works. Here is an algorithm that does work: choose $\lceil V/2 \rceil$ random vertices. You can calculate that each edge is chosen with probability at least $3/4$, and so the expected number of chosen edges is at least $3/4$ of all edges.

This algorithm can be derandomized (made deterministic) using the method of conditional expectations. I suggest you read about the method and work the details out. You will get something very similar to the greedy algorithm you're suggesting, but I suspect there will be some subtle differences.