C_j$ is an ordered linear chain for all $i \in \{n+1, n+2\}$ and $j \in \{n+3, n+4\}$.

$(3)$ For a divisor $D$ on $X$, we denote by $\#D$ the number of irreducible components of $D$.

The assertion(1) below follows from the fact that $C^2 = -2 - C . K_X \le -2$. The others are clear.

\[negdef\] $(1)$ If $C$ is a smooth rational curve on $X$, then $C^2 \le -2$.

$(2)$ If $\Gamma$ is of type $A_n$’, $D_n$’ or $E_n$’ then it is negative definite, i.e., the intersection matrix of components in $\Gamma$ is negative definite.

$(3)$ If $\Gamma$ is one of types $I_n^*$, $II^*$, $III^*$ and $IV^*$ (resp. $I_n^*$’, $II^*$’, $III^*$’ and $IV^*$’, but at least one component of $\Gamma$ is not a $(-2)$-curve), then $\Gamma$ is negative semi-definite (resp. negative definite).

$(4)$ If $\#\Gamma \le 5$, then $\Gamma$ is negative definite, unless $\Gamma$ supports a divisor of type $I_0^*$.

The Picard number can be estimated in the following way:

\[bound\] Suppose that the $(-2)$-components of $\Gamma$ do not support a divisor of type $I_0^*$. Let $r
= \min\{9, \, \#\Gamma - 1\}$. Then there is a subgraph $\Gamma'$ of $r$ components with negative definite intersection matrix. In 