Dynamic graph (?) - combination of connections between vertices that for each 3 exist min 1 edge
Your condition on the graph is the same as requiring its complement to be triangle-free. It is known that there are a lot of triangle-free graph. Indeed, since every bipartite graph is triangle-free, if we fix a partition of the vertex set into two sets of size $n/2$ (assuming for simplicity that $n$ is even), then we already get $2^{n^2/4}$ triangle-free graphs. It turns out that there aren't that much more, see this question on mathoverflow.

Perhaps you'd be interested in a paper on generating maximal triangle-free graphs up to isomorphism (in your case, you will be generating minimal graphs satisfying your condition), whose number is more manageable.