A construction of the generalized higher cluster category arising from an $(m+2)$-angulation of a marked surface

In this article, we study the $(m+2)$-angulations on a Riemann surface, characterized with its boundary components, punctures, and gender. We count the number of arcs in such a surface, and associate a graded quiver with superpotential associated with an $(m2)$-angulation. We show the compatibility between the flip of an $(m+2)$-angulation and the flip in the unpunctured case.