  Under a regularity condition for the singular point set, every flow on a
compact surface can be constructed by glueing four kinds of invariant open
subsets, which are flow boxes, transverse/periodic annuli, and locally dense
Q-sets. In particular, Morse-Smale flows on compact surfaces can be constructed
by glueing the boundaries of invariant flow boxes and invariant transverse
annuli, and Hamiltonian flows on compact surfaces with finitely many singular
points can be constructed by glueing the boundaries of invariant periodic
annuli. Using the decompositions, we construct a finite complete invariant for
surface flows of finite type. In fact, although degeneracy of singular points
implies uncountably many local topological equivalence classes and the set of
topological equivalence classes of minimal flows on a torus is uncountable, we
enumerate the set of topological equivalence classes of flows with at most
finitely many limit cycles but without non-closed recurrent orbits or
degenerate singular points on compact surfaces using finite labelled graphs.
