Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K 6 G, decides whether H ∩ K is finitely generated or not, and, in the affirmative case, it computes a set of generators for H ∩ K. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. F2×F2) even have unsolvable SIP.