Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the characterization of random reals as the set of reals in every full measure set coded by $W$. There are also theorems characterizing reals which cannot be added by forcing, for example the well known result that $0^\sharp$ cannot be added by forcing. What I want to know is if there are more general theorems characterizing (combinatorial, measure theoretic etc) properties of reals in $V$ guaranteeing that there is or is not a $\mathbb P \in W$ such that $x \in W^\mathbb P$?

I am particularly interested in this question with regards to properties related to cardinal characteristics of continuum. For instance, if there is a real $d \in V$ is dominating over the reals in $W$, then is there a nice way to characterize when $W[d]$ is (or is not) a generic extension of $W$?