Here is an incomplete answer, which I hope can be completed by others; therefore, I will make it CW, and invite others to add to it.

A groupoid is the path groupoid of a graph iff it is equivalent as a groupoid to a disjoint union of wedges of circles. Put another way, every groupoid is classified up to equivalence by a list (up to permutations) of groups, namely the fundamental groups of each component; a groupoid is the path groupoid of a graph iff every group on the list is a free group.

So one answer to your question might be: work out what all the fundamental groups are, and whether they are free.

This is not necessarily that easy. Without knowing more, I would expect that it is a very hard question of deciding whether some random group is free.