For any two graphs, a binary operation $\oplus$ is defined as follows:
And
Now, i call the operation $\oplus$ as merging of the graphs such that the identical vertices (if any) are stacked . But this name for the operation doesn't seem to be appropriate. So, what would be the standard\ appropriate name for this operation? Also, i am not clear what the 'term merging of graphs means?
Typically, we talk about equivalence classes of graphs - that is, the vertices are unlabelled, so we don't care about their names. In that case, this operation wouldn't make sense.
But if you want to talk about labelled graphs, I think the union would be the natural thing to call this: after all, the new vertex set is the union of the two component vertex sets, and the new edge set is the union of the two component edge sets. So I would write this as $G\cup H$, and I think most mathematicians would as well.