Let $\Gamma$ be a graph and $\bar{\Gamma}$ be its complement graph. I am looking for a relationship between the respective right-angled Coxeter groups $G(\Gamma)$ and $G(\bar{\Gamma})$.
My intuition is that, as $\Gamma$ and $\bar{\Gamma}$ together give the complete graph $K_n$, we should be able to somehow divide the relations of $\bar{\Gamma}$ out of $G(\Gamma)$ to get $G(K_n) = \mathbb{Z}_2^n$.
When $E_n$ is the empty graph, $G(E_n) = \ast^n (\mathbb{Z}_2)$. By the definition of the RACGs, we have the maps $\phi_1: G(E_n) \twoheadrightarrow G(\Gamma)$ and $\phi_2: G(E_n) \twoheadrightarrow G(\bar{\Gamma})$.
When dividing out the relations given by $\bar{\Gamma}$ from $G(\Gamma)$, we get the map $\rho_1: G(\Gamma) \twoheadrightarrow \mathbb{Z}_2^n$ and analogously $\rho_2: G(\Gamma) \twoheadrightarrow \mathbb{Z}_2^n$.
These maps should give a commutative square, $\rho_1 \circ \phi_1 = \rho_2 \circ \phi_2$. Is this square a pullback?
Also, can we relate $G(\Gamma)$ with $G(\bar{\Gamma})$ in some other way?
And what about when $\Gamma = \bar{\Gamma}$?