The quivers of surface type $A$ are constructed by taking a triangulation of a regular $(n+3)$-gon adding vertices to all edges of the polygon and adding arrows so that there is a clockwise-oriented triangle within each triangle of the triangulation. There is an example of this in Example 2.2, Figure 3 of Lauren William’s survey article. The $4$-valent vertices are precisely the vertices of the quiver corresponding to the diagonals of the triangulation. These quivers are known to be both restricted-mutation finite as well as produce finitely-many cluster variables.