Q:

To prove that the graph with a bijection function is connected

There is graph $(V,E)$, given that it is finite, connected undirected graph.
$W$ is a finite set and $f:V \rightarrow W$ is a bijective function.
I need to prove $(V \cup W,E \cup \{ (v, f(v))|  v \in V \})  $ is connected.
Can anyone tell what will be the set of edges here and how will bijection help in the connectivity.

A:

Let $G=\langle V,E\rangle$ be the original graph, and let $G'$ be the new graph. $G'$ has all of the edges that $G$ had. In addition, for each vertex $v\in V$ it has an edge between $v$ and $f(v)$. For example, if $G$ is a $4$-cycle, $G'$ looks like this:
                                  1'  
                                  |  
                                  1  
                                 / \  
                           2'---2   3---3'  
                                 \ /  
                                  4  
                                  |  
                                  4'

Here $V=\{1,2,3,4\}$, $W=\{1',2',3',4'\}$, and for each $v\in V$, $f(v)=v'$.

