Semi-edges, reflections and Coxeter groups

We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group:

Let $\pi : \Gamma \rightarrow (v,D,\iota,-1)$ be a $1$-covering of a monopole admitting semi-edges only. The graph $\Gamma$ is the Cayley graph of a Coxeter group if and only if $\pi$ is regular and any deck transformation in $\Delta(\pi)$ that interchanges two neighboring vertices of $\Gamma$ acts as a reflection on $\Gamma$.