For a fixed set $X$ and a finite collection $E_1,E_2,\ldots,E_k\subseteq X$, define the binary relation adjacency as follows: $E_i,E_j$ are adjacent if their intersection is nonempty. We term the transitive closure of this relation by transitive adjacency and define the adjacent union by $$ \tilde\cup(E_1,\ldots,E_k):= \begin{cases} \bigcup_{i=1}^k E_i, & \text{the $(E_i)$ are transitively adjacent} \\ \emptyset, & \text{else} . \end{cases} $$

Are there standard terms for transitive adjacency and adjacent union?