1$ or $2$
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Throughout this section, we assume that $X$ is a nonsingular projective surface with $K_X$ nef and Kodaira dimension $\kappa(X)
= 1$ or $2$. The main result is Proposition \[key\] to be used in the next section.

Up to Lemma \[bound\], we let $\Gamma$ be a connected effective integral divisor on $X$ which consists of smooth rational curves and has a (rational) tree as its dual graph.

$(1)$ We say that $\Gamma$ is of type $A_n'$ (resp. $D_n'$, or $E_n'$) if its weighted dual graph is of Dynkin type $A_n$ (resp. $D_n$, or $E_n$) but its weights may not all be $(-2)$.

$(2)$ $\Gamma$ is of type $I_n^*$ (resp. $II^*$, or $III^*$, or $IV^*$) if $\Gamma$ is of the respective elliptic fibre type (hence ${\text{\rm Supp}}(\Gamma)$ is a union of $(-2)$-curves). $\Gamma$ is of type $I_n^*$’ (resp. $II^*$’, or $III^*$’, or $IV^*$’) if $\Gamma$ is equal to an elliptic fibre of type $I_n^*$ (resp. $II^*$, or $III^*$, or $IV^*$), including coefficients, but the self intersections of components of $\Gamma$ may not all be $(-2)$. E.g. $\Gamma = 2\sum_{i=0}^n C_i + \sum_{j=n+1}^{n+4} C_j$ is of type $I_n^*$’, where $C_i + C_0 + C_1 + \dots + C_n + 