fundamental groupoids for graphs

in recent years several notions of discrete homotopy for graphs have been introduced, including a notion of ×-homotopy due to dochtermann. in this paper, we define a ×-homotopy fundamental groupoid for graphs, and prove that it is a functorial ×-homotopy invariant for finite graphs. we also introduce tools to compute this fundamental groupoid, including a van kampen theorem. we conclude with a comparison with previous definitions along these lines, including those built on polyhedral complexes of graph morphisms.