  The stringy Euler number and E-function of Batyrev for log terminal
singularities can in dimension 2 also be considered for a normal surface
singularity with all log discrepancies nonzero in its minimal log resolution.
Here we obtain a structure theorem for resolution graphs with respect to log
discrepancies, implying that these stringy invariants can be defined in a
natural way, even when some log discrepancies are zero, and more precisely for
all normal surface singularities which are not log canonical. We also show that
the stringy E-functions of log terminal surface singularities are polynomials
(with rational powers) with nonnegative coefficients, yielding well defined
(rationally graded) stringy Hodge numbers.
