  We construct a complete invariant for non-wandering surface flows with
finitely many singular points but without locally dense orbits. Precisely, we
show that a flow $v$ with finitely many singular points on a compact connected
surface $S$ is a non-wandering flow without locally dense orbits if and only if
$S/v_{\mathrm{ex}}$ is a non-trivial embedded multi-graph, where the extended
orbit space $S/v_{\mathrm{ex}}$ is the quotient space defined by $x \sim y$ if
they belong to either a same orbit or a same multi-saddle connection. Moreover,
collapsing edges of the non-trivial embedded multi-graph $S/v_{\mathrm{ex}}$
into singletons, the quotient space $(S/v_{\mathrm{ex}})/\sim_E$ is an abstract
multi-graph with the Alexandroff topology with respect to the specialization
order. Therefore the non-wandering flow $v$ with finitely many singular points
but without locally dense orbits can be reconstruct by finite combinatorial
structures, which are the multi-saddle connection diagram and the abstract
multi-graph $(S/v_{\mathrm{ex}})/\sim_E$ with labels. Moreover, though the set
of topological equivalent classes of irrational rotations (i.e. minimal flows)
on a torus is uncountable, the set of topological equivalent classes of
non-wandering flows with finitely many singular points but without locally
dense orbits on compact surfaces is enumerable by combinatorial structures
algorithmically.
