Generic immersions of the two-sphere to $\mathbf R^3$ and their skeleta

Let $f\colon S^2\looparrowright\mathbb R^3$ be a generic smooth immersion. The skeleton of $f$ is the following triple $(\Gamma, D, p)$. $\Gamma$ is the 1-polyhedron of singular points of $f$, $D=f^{-1}(\Gamma)$ is also a 1-polyhedron, and $p\colon D\to\Gamma$, $x\mapsto f(x)$, is the projection. For triples of the form $(D,\Gamma, p)$, where $\Gamma$ has at most 4 vertices, we give an iff-condition under which the triple is the skeleton of a smooth immersion $f\colon S^2\looparrowright\mathbb R^3$.