  The conformal invariance and universality results of Chelkak-Smirnov on the
two-dimensional Ising model hold for isoradial planar graphs with critical
weights. Motivated by the problem of extending these results to a wider class
of graphs, we define a generalized notion of s-holomorphicity for functions on
arbitrary weighted surface graphs. We then give three criteria for
s-holomorphicity involving the Kac-Ward, Kasteleyn, and discrete Dirac
operators, respectively. Also, we show that some crucial results known to hold
in the planar isoradial case extend to this general setting: in particular,
spin-Ising fermionic observables are s-holomorphic, and it is possible to
define a discrete version of the integral of the square of an s-holomorphic
function. Along the way, we obtain a duality result for Kac-Ward determinants
on arbitrary weighted surface graphs.
