Smallest vertex cover which is also an independent set
This is the "Independent Vertex Cover" problem. It is solvable in polynomial time. To see this, note that for every edge, exactly one endpoint of the edge must be in a vertex cover. We can reduce the problem to 2-SAT, as follows:  make a variable $x_i$ for each vertex $i$, and for each edge $(i,j)$, include clauses of length two of the form $x_i \oplus x_j = 1$. 

See also https://stackoverflow.com/questions/10769971/algorithm-for-independent-vertex-cover