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