TITLE: $2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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, Invent. Math. 4, 336-358 (1968). ZBL0219.14003.