Graph Theory: Isomorphic graphs

Question

Show that the inverse of an isomorphism of graphs is also an isomorphism of graphs.

So, I just started a graph theory course and am having a little trouble with one of the problems on the homework. I know that a graph is isomorphic if there are bijections $Θ:V(G)→V(H)$ and $Φ:E(G)→E(H)$ such that $Ϯ_G(e)=uv$ if and only if $Ϯ_H(Φ(e))=Θ(u)Θ(v)$. That is, they have the same structure, but differ only in the names of the vertices and edges. I just don't know how to find the inverse of an isomorphic graph.

Answer 1

Since $\Theta$ and $\Phi$ are bijections, thus they have inverses $\Theta^{-1}$ and $\Phi^{-1}$.

Now let $e$ denote an edge of $H$ and $u,v$ denote vertexes. All that remains is to show the following equivalence. $$I_H(e)=uv \iff I_G(\Phi^{−1}e)=(\Theta^{-1}u)(\Theta^{-1}v).$$

We proceed by a sequence of equivalences.

1. $I_H(e)=uv$
2. $I_H(\Phi \Phi^{-1}e) = (\Theta\Theta^{-1}u)(\Theta\Theta^{-1}v)$
3. $I_G(\Phi^{-1}e) = (\Theta^{-1}u)(\Theta^{-1}v)$

A bit of explanation: we got from Line 1 to Line 2 by using the very definition of inverse functions, whereas we got from 2 to 3 by using the equivalence (the iff) given in the question.