I'm interested if the following language construction was studied. Let $\Sigma$ be an alphabet and $\Sigma^*$ is the set of all words over $\Sigma$. Consider a directed graph $G=\langle\Sigma^*,E\rangle$ together with a subset $S\subseteq\Sigma^*$ of its vertices. Now, define the language $L=\langle\Sigma,E,S\rangle$ consisting of all words reachable from $S$ in $G$. That is if $w\in L$ then $w\in S$ or there is a word $u\in S$ and some path from $u$ to $w$ in $G$. Can we describe $L$ using the Chomsky hierarchy?