** **

**Abstract:**

On
a Riemann surface $F$ (with or without boundary)

let
a finite number of points $P_i$ with tangent vectors $V_i$ be

specified.
Choosing a type $\nu_i = 0, 1, 2, \ldots $ and a

positive
real coefficient $A_i$ for each point,

this
determines a unique real-valued harmonic function

$u
: F \to \mathbb{R} \cup \infty$.

The
gradient flow of $u$ gives a graph $\mathcal{K}$, with the above

points
$P_i$ and the critical points as vertices and flow lines

and
level curves as edges.

The
complement of that graph is a tesselation of the surface $F$.

(Using
that $u$ and the graph varies continuously with the complex

structure,
this gives a cell decomposition of the moduli space.)