Don't understand this graph definition
D.W.'s advice to consider some small examples is excellent.  Two great examples to start with are an isolated vertex and a single edge!  Once you can how the construction transforms vertices and edges, you know what happens to the building blocks.

Also, some familiarity with the basics of these constructions helps.  In particular, any time you see "Let the vertex set of the new graph be $V\times S\,$" for some set&nbsp;$S$, you should think: "Take a copy of&nbsp;$V$ for every element of&nbsp;$S$, and label each copy with the element it's associated with."  So, here, the vertex set of&nbsp;$H$ is two copies of $G$'s vertex set: one labelled&nbsp;$0$ and one labelled&nbsp;$1$.  Now, look at the edges.  The first set of edges says, "For every vertex $v\in G$, add an edge from copy&nbsp;$0$ to copy&nbsp;$1$." The second set says, "For every edge $(x,y)\in G$, add an edge from copy&nbsp;$1$ of&nbsp;$x$ to copy&nbsp;$0$ of&nbsp;$y$."