Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups

There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (Γ) and a subgraph (Λ) of its complement, checking some conditions, and, if satisfied, producing a graph (Δ) such that the right-angled Artin group with presentation graph (Δ) is a finite index subgroup of the right-angled Coxeter group with presentation graph (Γ). They do not tell us how to find (Λ), given (Γ). We show, in the 2–dimensional case, that the existence of such a (Λ) is connected to the graph property of satellite-dismantlabilty of (Γ), and we use this to give an algorithm for producing a suitable (Λ) or deciding that one does not exist.