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In the study of groups acting on graphs, one often starts with hypothesis "Let $X$ be a graph, and $Aut(X)$ acting on $X$ without inversion.."

But, is it true that $Aut(X)$ always acts on $X$ without inversion?

Here $Aut(X)$ is the set of bijections between all vertices of $X$ which preserve adjcency, and a group $G$ acts without inversion means there is no oriented edge $e$ of $X$ such that $g.e = \bar{e}$, $g\in G$; $\bar{e}$ is the edge $e$ with reverse orientation.

Some people say that, "if $G$ is acting on $X$ with inversion, then taking barycentric subdivision, we will have action of group $G$ without inversion.."

But, by taking barycentric division of $X$, is it not true that its automorphism group may also change?

What would be the precise way to consider the action of full automorphism group on the graph and study its quotient graph?

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    Taking $X$ to be the connected tree on two vertices shows that $\operatorname{Aut}(X)$ does not necesarily act without inversions... I guess you knew this?2011-07-19

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