Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let $\pi: X \to \text{Spec}(R)$ be minimal resolution of singularities with exceptional divisor $E=\pi^{-1}(\mathfrak m)$. If $E$ has exactly one irreducible component, then must it be true that $R$ is a [cyclic quotient singularity](https://www.jstage.jst.go.jp/article/kyotoms1969/10/1/10_1_293/_article/-char/en)?



$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
The answer is *yes*, at least over $\mathbb{C}$, since $2$-dimensional (cyclic) quotient singularities are *taut* (*starr*, in German), namely, they are uniquely characterized, up to biholomorphisms, by their resolution graph.


In other words, every $2$-dimensional normal singularity, having the same resolution graph of a (cyclic) quotient singularity, is itself a (cyclic) quotient singularity.


See Korollar 2.12 in


E. Brieskorn: [Rationale Singularitäten komplexer Flächen](http://dx.doi.org/10.1007/BF01425318), *Invent. Math.* **4**, 336-358 (1968). [ZBL0219.14003](https://zbmath.org/?q=an:0219.14003).


