  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.
