Irreducible Contact Curves via Graph Stratification

We prove that the moduli space of contact stable maps to ${\mathbb{P}}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in ${\mathbb{P}}^{2n+1}$ with any Schubert condition. We give explicitely some of these invariants for ${\mathbb{P}}^3$ and ${\mathbb{P}}^5$. We give another proof of the formula for the number of plane contact curves in ${\mathbb{P}}^3$ meeting the appropriate number of lines.