Classification of weighted dual graphs with only complete intersection singularities structures

Let $p$ be normal singularity of the 2-dimensional Stein space $V$. Let $\pi\colon M\to V$ be a minimal good resolution of $V$, such that the irreducible components $A_i$ of $A=\pi^{-1}(p)$ are nonsingular and have only normal crossings. Associated to $A$ is weighted dual graph $\Gamma$ which, along with the genera of the $A_i$, fully describes the topology and differentiable structure of $A$ and the topological and differentiable nature of the embedding of $A$ in $M$. In this paper we give the complete classification of weighted dual graphs which have only complete intersection singularities but no hypersurface singularities associated to them. We also give the complete classification of weighted dual graphs which have only complete intersection singularities associated with them.