If you are only considering monomial algebras (that is, if you are generating the ideal I by paths) then your intuition about overlaps is correct. There is a paper by Bardzell (The alternating behaviour of monomial algebras) where he constructs explicitely a projective resolution of the quotient algebra as a bimodule over itself (whose length bounds the gldim of the algebra) which is constructed precisely by considering overlaps.
By the way, if the graph is not acyclic, then the global dimension can very well be infinite. The simplest example is a quiver with one vertex and a loop, and the ideal geberated by the square of the loop.