Can the group generated by local complementations, ${lc_i|i=1,\cdots,n}$  on simple graphs on $n$ vertices, be categorized as a coxeter group? After all these obey:
\begin{equation}
\langle lc_i| (lc_i lc_j)^{m_{ij}}=1\rangle \quad where \quad 
m_{ij}=\{ 2,3,6\}
\end{equation}

If so, which classification do they belong to?

A few definitions:

Local complementation, $lc_k$, of vertex $k$ replaces the subgraph induced on the neighborhood, $N_k$ of $k$, by its complement. The transformation on the adjacency matrix, $A_G$ of a graph, $G$ is:

$lc_k: (A_G)_{ij} \rightarrow (A_G)_{ij}+(A_G)_{ik}(A_G)_{jk}+Diagonal[(A_G)_{ij}+(A_G)_{ik}(A_G)_{jk}]$.

Where the addition is $mod(2)$.

We did some numerical work in this regard, see my question "the group of local complementation on simple graphs". We find an interesting pattern which leads us to believe there must be an (if not trivial) interesting underlying group structure.