I have a homework problem where I have a graph $G$ and I am tasked with proving that at least one of $G$ and $G$ complement is connected. However, I am unclear on the exact meaning of $G$ complement. 

For example, let's imagine I have a disconnected graph with four vertices $(V\_1, V\_2, V\_3, \text{and } V\_4)$. 
If the edges form a sort of box where the bottom edge is left disconnected, would $G$ complement have that edge filled in along with the cross edges as well? Furthermore, does $G$ complement contain all of the edges in $G$ or just the edges not contained in $G$? Thanks for taking the time to read.