On Matsushita ${\pi }_1^2$ discrete fundamental groups

The Matsushita fundamental groups of a graph $X$, denoted ${\pi }_1^r(X)$, are certain discrete versions of the fundamental group for topological spaces. For $r=2$, these groups have a nice combinatorial description, due to Sankar. In this paper we prove two results about ${\pi }_1^2$. First, we prove a Seifert-van Kampen-type theorem. Similar results have previously been obtained by Barcelo, et al. (and strengthened by Kapulkin and Mavinkurve) for a different notion of discrete fundamental group. Second, we prove that an arbitrary group $G$ can be realized as ${\pi }_1^2(X)$ for some graph $X$. Our construction works equally well for the aforementioned alternate discrete fundamental group $A_1(X)$, and our second result thus also provides an entirely different method of proof for a theorem of Kapulkin and Mavinkurve.