We say that $E,E'\subset X$ are *adjacent*
if $E\cap E'\neq\emptyset$. Adjacency is an equivalence relation
over nonempty subsets of $X$ and we term its transitive closure by
*transitive adjacency*.
For $E_1,\ldots,E_k\subset X$
define their *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*?