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}=\begin{cases} 2,3\end{cases}
\end{equation}

If so, which classification do they belong to?