Let H be a hexagonal system. The z-transformation graph Z(H) of H is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric differences is a hexagon of H. In “z-transformation graphs of perfect matchings of hexagonal systems” (to appear) by F. Zhang, X. Guo, and R. Chen, it was proved that the connectivity \(\kappa\) (Z(H)) of Z(H) is equal to \(\delta\) (Z(H)). Furthermore, in this paper we determine a class \({\mathcal H}\) of hexagonal systems with \(\kappa (Z(H))=1\) and subclass \({\mathcal H}_ 2\) of \({\mathcal H}\) such that every hexagonal system in \({\mathcal H}_ 2\) has exactly two vertices of degree one. Finally the enumeration problem of \({\mathcal H}_ 2\) is considered.