Decomposition of surface into quadrilaterals

There are many approaches to the classification of Morse functions and their gradient fields (Morse Fields) on 2-surfaces. This paper studies the gluings of quadrilaterals and the classification of topological surfaces obtained by gluing together sides of quadrilateral graphs created by trajectories on gradient fields with focal critical points as vertices and a saddle point in center for classification saddle and focus critical points. We review the graphs embedded into surfaces associated with gluings, the number of possible gluings, and some algorithms based on labeling schemes of fundamental polygons that can be used to compute those numbers.