Graphs of Stable Maps from 3-Manifolds to 3-space

We associate a graph with weights in its vertices and edges to any stable map from a 3-manifold to . These graphs are -invariants from a global viewpoint. We study their properties and give a sufficient and necessary condition for a graph to be the graph of a stable map from a 3-sphere with handles to . We also obtain a sufficient condition in the general case of a closed stably paralellizable 3-manifold in terms of its Heegaard genus.