Given a graph $G$ and two of its subgraphs $A$ and $B$ we define another subgraph, $A+B\subseteq G$ as the subgraph with the following properties,
- The vertex set of $A+B$ is the union of the vertex sets of $A$ and $B$
- The edge set of $A+B$ is the union of the edge sets of $A$ and $B$ but also includes any edges joining $A$ and $B$ that are also edges in $G$, i.e. if a $a\in V(A)$ and $b\in V(B)$ then $\{a,b\}$ is an edge of $A+B$ if and only if $\{a,b\}$ is an edge in $G$.
Does this operation on subgraphs have a name? In the case where $A$ and $B$ are disjoint subgraphs of $K_n$ then this would be equivalent to the graph join, hence my first guess for a name would be the subgraph join.