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We introduce a graph associated with any stable map defined from the connected sum # n (S 1 × S 2) of n copies of the product S 1 × S 2 to the Euclidean 3-space. This graph has a weight on each vertex and a pair of weights on each edge, and its properties provide a necessary and sufficient condition to be the graph of a stable map defined from # n (S 1 × S 2) to the Euclidean 3-space. [ABSTRACT FROM AUTHOR]

In this paper, we define the concept of graph extension, embedded on a closed and orientable surface, associated to a 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 . [ABSTRACT FROM AUTHOR]

In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps from S 3 to R 3 , the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets. [ABSTRACT FROM AUTHOR]