Graph based topological analysis of tessellated surfaces

In this paper a graph-based method is presented which not only characterizes topological classification of the tessellated surfaces but also simultaneously generates the substantial circles or generators on the surface. Canonical polygons cannot always be mapped back to the original surface in terms of the edges of the given triangles. Hence, instead of applying canonical transformation to the initial "word", an associated graph is constructed using the unique vertices in the word. The graph is then decomposed into its constituent loops and paths. Based on the type of edges present, the loops are classified into three types. The number of loops of each type in the graph is then used for counting the rank or genus and classification of the given surface as being open or closed, orientable or non-orientable. The image of the loops and paths on the original surface give the substantial circles and arcs on the surface respectively.