  The perfect quantum strategies of a linear system game correspond to certain
representations of its solution group. We study the solution groups of graph
incidence games, which are linear system games in which the underlying linear
system is the incidence system of a (non-properly) two-coloured graph. While it
is undecidable to determine whether a general linear system game has a perfect
quantum strategy, for graph incidence games this problem is solved by
Arkhipov's theorem, which states that the graph incidence game of a connected
graph has a perfect quantum strategy if and only if it either has a perfect
classical strategy, or the graph is nonplanar. Arkhipov's criterion can be
rephrased as a forbidden minor condition on connected two-coloured graphs. We
extend Arkhipov's theorem by showing that, for graph incidence games of
connected two-coloured graphs, every quotient closed property of the solution
group has a forbidden minor characterization. We rederive Arkhipov's theorem
from the group theoretic point of view, and then find the forbidden minors for
two new properties: finiteness and abelianness. Our methods are entirely
combinatorial, and finding the forbidden minors for other quotient closed
properties seems to be an interesting combinatorial problem.
