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.

  • 3
    $\begingroup$ Hint: Since $\Theta$ and $\Phi$ are bijections, thus they have inverses $\Theta^{-1}$ and $\Phi^{-1}$. So you just have to show that $I_H(e) = uv$ iff $I_G(\Phi^{-1}(e)) = \Theta^{-1}(u)\Theta^{-1}(v).$ Also, you're not trying to find the inverse of an "isomorphic graph," so much as an inverse of a graph isomorphism. $\endgroup$ Jan 13 '14 at 21:19
  • $\begingroup$ So since Θ and Φ are bijections, they have inverses Θ^(−1) and Φ^(−1). But how do I show that I_H(e)=uv iff I_G(Φ^(−1)(e))=Θ^(−1)(u)Θ^(−1)(v)? This is my first homework so if you could just explain it to me, I would really appreciate that. Thanks $\endgroup$
    – Brian
    Jan 13 '14 at 22:05
  • $\begingroup$ Sure, I just wrote up an answer. Its not especially well-written though, so I suggest you put it into your own (probably better) words before handing it in. $\endgroup$ Jan 13 '14 at 22:19
  • $\begingroup$ Thank you, that helped a lot. $\endgroup$
    – Brian
    Jan 13 '14 at 23:03

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.


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