Complements of graphs of meromorphic functions and complete vector fields

Given a meromorphic function $s: \mathbb{C}\to \mathbb{P}^{1}$, we obtain a family of fiber-preserving dominating holomorphic maps from $\mathbb{C}^{2}$ onto $\mathbb{C}^{2}\setminus graph(s)$ defined in terms of the flows of complete vector fields of type $\mathbb{C}^{\ast}$ and of an entire function $h:\mathbb{C}\to\mathbb{C}$ whose graph does not meet $graph(s)$, which was determined by Buzzard and Lu. In particular, we prove that the dominating map constructed by these authors to prove the dominability of $\mathbb{C}^{2}\setminus graph(s)$ is in the above family. We also study the complement of a double section in $\mathbb{C}\times\mathbb{P}^{1}$ in terms of a complex flow. Moreover, when $s$ has at most one pole, we prove that there are infinitely many complete vector fields tangent to $graph(s)$, describing explicit families of them with all their trajectories proper and of the same type ($\mathbb{C}$ or $\mathbb{C}^{\ast}$), if $graph(s)$ does not contain zeros; and families with almost all trajectories non-proper and of type $\mathbb{C}$, or of type $\mathbb{C}^{\ast}$, if $graph(s)$ contains zeros. We also study the dominability of $\mathbb{C}^{2}\setminus A$ when $A\subset \mathbb{C}^{2}$ is invariant by the flow of a complete holomorphic vector field.