$C_1$ nor $C_4$ if we choose the two points to be general. Then $C_3$ (resp. $C_2$) has to connect both $C_1$ and $C_4$. Then by Lemma \[lem:smoothable\], there is a very free curve with smaller $-K_X$ degree, which is a contradiction.

Thus the second case does not contribute to the Gromov-Witten invariant.

If $C_2$ (resp. $C_3$) deforms in a divisor, then it cannot pass through two general points in the divisor. In particular, it cannot meet $\Gamma$ and $C_1$ (or $C_4$).

Without loss of generality, assume $C_2$ is connected to $C_1$. If $C_2$ can pass through two general points in the divisor, then it can connect both $C_1$ and a general deformation of $C_1$ (Note $C_1$ has positive intersection with the divisor). Then there is a very free curve with smaller $-K_X$ degree by Lemma \[lem:smoothable\]. Again a contradiction.

Thus the third case has no contributions either.

In the fourth case, assume the curve $C_2$ deforms to a free curve. In particular, $C_2$ lies in a component of the Kontsevich moduli space of dimension $3$. Meeting a curve is a codimension $2$ condition. So if we choose the points and the curve $\Gamma$ to be general enough, $C_2$ cannot meet 