Recent advances in the parameterization and adaptive sampling of disc-like surfaces have brought a renewed interest on the global parameterization problem and, more specifically, on the cut-graph search. This paper focuses on the calculation of a family of generators and cut-graphs for the global parameterization of arbitrary triangle meshes. This result is achieved by combining the construction of harmonic scalar fields f : M rarr R of known maxima and minima with the quasi Morse-Smale complex of(M, f). The proposed technique has a simple implementation and outperforms previous work in terms of smoothness of the cut-graphs, stability with respect to the surface sampling, tessellation, topological noise (e.g., tiny handles), and capability of handling boundary components. Since we generate a family of cut-graphs, we also provide a comparison between the parameterizations of M induced by two cut-graphs.