  An exposition on Spivakovsky's dual graphs of valuations on function fields
of dimension two is first given, leading to a proof of minimal generating
sequences for the non-divisorial valuations. It should be noted that the
definition of generating sequence used in this paper is different from
Spivakovsky's original usage. This change leads to an explicit formulation of
generating sequence values for the non-divisorial cases in terms of data from
their dual graphs. The proofs are elementary in the sense that only continued
fractions and the linear Diophantine Frobenius problem from classical number
theory are used.
