Morphisms between right-angled Coxeter groups and the embedding problem in dimension two

In this article, given two finite simplicial graphs $\Gamma_1$ and $\Gamma_2$, we state and prove a complete description of the possible morphisms $C(\Gamma_1) \to C(\Gamma_2)$ between the right-angled Coxeter groups $C(\Gamma_1)$ and $C(\Gamma_2)$. As an application, assuming that $\Gamma_2$ is triangle-free, we show that, if $C(\Gamma_1)$ is isomorphic to a subgroup of $C(\Gamma_2)$, then the ball of radius $8|\Gamma_1||\Gamma_2|$ in $C(\Gamma_2)$ contains the basis of a subgroup isomorphic to $C(\Gamma_1)$. This provides an algorithm determining whether or not, among two given two-dimensional right-angled Coxeter groups, one is isomorphic to a subgroup of the other.