Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.

My question is:

what are the conditions on $X$ so that $\cal X_{s}$ does not contain any rational curves?