Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow.

I'm trying to get some intuition about how much the global dimension can grow when we quotient by some homogeneous ideal of relations I. In general, if Q is acyclic (is this necessary?), then the global dimension is bounded by the number of vertices, but I want something that uses information from I. For simplicity, let's assume that there is at most 1 edge between any two vertices. If I add the relation that a single path of length 2 is 0, then the global dimension goes up to 2, and the same is true if 2 is replaced by any r>2 (right?). I can get higher global dimensions by the following: take some consecutive arrows $a_1, a_2, \dots, a_n$, and require that each path of length 2 $a_{i+1} a_i$ is 0, then the global dimension goes up to n-1.

The way I am trying to picture this is by thinking of projective modules as flowing water which gets stopped by some rock placed where the relations are, and seeing how many times the flow needs to restart before it can reach the end (I don't know if this is a useful comment.)

Anyway, here is my question: is there some simple way to bound the global dimension of Q/I assuming that Q is acyclic and no multiple edges between any two vertices? In this case, we're only allowed to say that certain paths are 0, so I am suspecting this has something to do with "number of overlaps." My guess would be something like, define an overlap to be when an initial segment of one path coincides with an ending segment of another path, and then the global dimension should be less than or equal to number of overlaps + 2.