Characterize which superquivers arise as Hom-Ext quivers

Determine whether a superquiver is a Hom-Ext quiver of some algebra if and only if it admits at least one irreducible representation in a suitable hereditary triangulated category.

Background

The paper introduces superquivers as quivers with relations whose arrows have degree zero or one, and interprets exceptional collections as representations of such superquivers in triangulated categories. However, the authors emphasize that not every superquiver representation is necessarily realized by a Hom-Ext quiver.

They propose irreducibility, defined through the inner graded endomorphism algebra and the radical-square condition, as a necessary property for realizability. The unresolved question is whether this necessary condition is also sufficient.