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

Abstract 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-dismantlability of Γ, and we use this connection to give an algorithm for producing a suitable Λ or deciding that one does not exist.