Given a basic split finite dimensional algebra $A$ over a field K, A is isomorphic to $KQ/I_1$, for some quiver Q and a minimal(meaning it is generated by relations $x_i$, such that no relation is implied by any other) admissible ideal $I_1$. Now A can be isomorphic to $KQ/I_2$ for a different minimal admissible ideal $I_2$. There are examples where $I_1$ is monomial but $I_2$ is not.
Question: Given $A=KQ/I_1$, how can one know wheter there exists a monomial minimal admissible ideal $I_2$ such that $A \cong K/I_2$?
For example, I think if $A$ is a selfinjective algebra then this can never happen expect when $A$ is a Nakayama algebra.