Another vertex cover question? I'm not sure this is equivalent to bipartite vertex cover question. The question is:

Given a BIPARTITE graph, what is the minimum number of vertex from the right side whose edges cover all vertex from the left side. 

e.g. In the following graph, the answer is 1, cause vertex g has connection to all vertex on the left.
It is equivalent to the set cover problem. You can regard each vertex in the right side as a set of its neighbors in the left side. In your example, $e,f,g$ correspond respectively to the sets $\{d\},\{d\},\{a,b,c,d\}$. Now a minimum vertex cover in your problem is equivalent to a minimum set cover.