  We take a graph theoretic approach to the problem of finding generators for
those prime ideals of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ which are
invariant under the torus action ($\mathbb{K}^*)^{m+n}$. Launois
\cite{launois3} has shown that the generators consist of certain quantum minors
of the matrix of canonical generators of
$\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ and in \cite{launois2} gives an
algorithm to find them. In this paper we modify a classic result of
Lindstr\"{o}m \cite{lind} and Gessel-Viennot~\cite{gv} to show that a quantum
minor is in the generating set for a particular ideal if and only if we can
find a particular set of vertex-disjoint directed paths in an associated
directed graph.
