Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve

Let $(V, p)$ be a normal surface singularity. Let $\pi\colon (M, A)\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\Gamma$ associated to $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is $-3$-curve, and the rest are all $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see Section 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give the formulas for computing arithmetic and geometric genera of star-shaped graphs.
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