2. Universidad Cardenal Herrera-CEU

In this paper, we de ne the concept of graph extension, embedded on a closed and orientable surfaces, associated to pairing of edges of regular polygons in order to show that the K-regular pairing of edges graphs can be obtained by the canonical extension of graphs (graphs with a single vertex). We will present examples of K-regular graphs associated to surfaces with genus g < 3.

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (i.e. fold maps) from closed surfaces to the projective plane.