Constructing presentations of subgroups of right-angled Artin groups

Let be the right-angled Artin group associated to the flag complex and let be its canonical height function. We construct an algorithm that, given and for , outputs a presentation for of optimal deficiency on a minimal generating set, provided is triangle-free; the deficiency tends to infinity as if and only if the corresponding Bestvina-Brady kernel is not finitely presented, and the algorithm detects whether this is the case. We explain why there cannot exist an algorithm that constructs finite presentations with these properties in the absence of the triangle-free hypothesis. We also prove, for general , that the abelianized deficiency of tends to infinity if and only if is -acyclic, and discuss connections with the relation gap problem.