Induced universal graphs for families of small graphs

For $0 \leq k \leq 6$, we give the minimum number of vertices $f(k)$ in a graph containing all $k$-vertex graphs as induced subgraphs, and show that $16 \leq f(7) \leq 18$. For $0 \leq k \leq 5$, we also give the counts of such graphs, as generated by brute-force computer search. We give additional results for small graphs containing all trees on $k$ vertices.