Is this greedy algorithm for vertex cover studied before?
Unfortunately, the algorithm can be arbitrarily bad.  In the following example, each vertex $u_i$ has $d$ disjoint neighbors, of which only four are drawn. The optimal solution is $\{u_1, \ldots, u_p, w_1, \ldots, w_d\}$, while the greedy algorithm returns its complement.  The ratio is $\frac{d p}{p + d} \approx d$ with $p\to 0$.