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