  It is known that a two-dimensional $F$-rational ring has a rational
singularity. However a two-dimensional ring with a rational singularity is not
$F$-rational in general. In this paper, we investigate $F$-rationality of a
two-dimensional graded ring with a rational singularity in terms of the
multiplicity. Moreover, we determine when a two-dimensional graded ring with a
rational singularity and a small multiplicity is $F$-rational.
