Q:

Union, intersection and difference operations with cycle graphs

I am confused with some graph operations, so would like to clarify some simple questions;
If I have below sub graphs (cycles) such as $G_1$, $G_2$, $G_3$ & $G_4$

and if i want to do the union, intersection and difference operation of each graph after the other in the following way

then, I assume the graphs that i have drawn are correctly representing the results of $G_5$, $G_6$ and $G_7$. 

So my question is; Do these figures correctly represent the results?
If not, what type of operations should I use to get graphs shown above?

A:

The way you've denoted graphs, e.g. $G_1=\{1,2,3,1\}$, I would describe as incorrect.  Aside from that, what you write seems correct.  I would describe $G_1,\ldots,G_4$ as follows:

$G_1$ is the cycle induced by the vertices $\{1,2,3\}$,
$G_2$ is the cycle induced by the vertices $\{1,3,4,5\}$,
$G_3$ is the subgraph induced by the vertices $\{3,6\}$, and
$G_4$ is the cycle induced by the vertices $\{6,7,8\}$.

I would describe $G_5$ as a symmetric difference (which is sometimes denoted $\bigtriangleup$) of graphs.

$G_5=G_1 \bigtriangleup G_2$.

And since $G_5 \cap G_3$ and $G_6 \cap G_4$ have no edges, we can define

$G_6=G_5 \cup G_3$, and
$G_7=G_6 \cup G_4$.

Although, presumably this operation is going to be used outside of this toy example, in which case $G_6=G_5 \bigtriangleup G_3$ and $G_7=G_6 \bigtriangleup G_4$ are also correct.

