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Its the lexicographic product of the empty graph on two vertices by your original graph. (The terminology for lexicographic product is a bit confused - it's not commutative and there are two schools of thought about which graph should go first.)

What you asking for is currently not possible. When you edit a the output of Plot in the manner you describe, you make the plot graphic independent, i.e., no longer related to the Plot expression (note the cell is no longer grouped with the code). Such an independent graphic is like one created with the drawing tools palette. Some operations with these tools can be undone, others not. To see if the last operation you performed can be undone, check whether or not Edit > Undo is enabled (black rather than grayed out). In the case of moving a curve from a plot graphic, Undo will be grayed out.