On BD-algebra and CM-Auslander algebra for a gentle algebra and their representation types

Let $A$ be a gentle algebra, and $B$ and $C$ be its BD-gentle algebra and CM-Auslander algebra, respectively. In this paper, we show that the representation-finiteness of $A$, $B$ and $C$ coincide and the representation-discreteness of $A$, $B$ and $C$ coincide.