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Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. This induces a graph morphism $F$ from the graph $G$ of $P$ to the graph $H$ of $Q$: take the vertex $A$ in $G$, viewed as a circle in $P$, to the circle $f(A)$; take the edge $p$ in $G$, viewed as the intersection of distinct circles in $P$, to the point $f(p)$ if the endpoints $A$ and $B$ of $p$ are mapped to distinct vertices, and take $f(p)$ to $f(A)$ otherwise.

In Harmonic morphisms and hyperelliptic graphs, Baker & Norine define a notion of a harmonic morphism between graphs (assumed to be finite and without loops, although multiple edges are allowed): A graph morphism $\phi\colon G \to H$ is harmonic if for all vertices $x\in G$, the quantity $$|\{ e\in \phi^{-1}(e'): e\text{ is incident to }x \}|$$ is independent of choice of edge $e'$ incident to $\phi(x)$.

Returning to the morphism $F$ defined above, and restricting $P$ and $Q$ to each have finitely many circles, under what conditions is $F$ a harmonic morphism?

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I don't understand why there exists a conformal map f taking circles to circles. Can you give at least one non-trivial example of such map? –  Alexandre Eremenko Mar 9 '13 at 23:48
    
@AlexandreEremenko I'm not sure. I was just guessing there would be. –  Avi Steiner Mar 10 '13 at 0:59
    
@AlexandreEremenko probably the OP was supposing that the contact graphs of P and Q are triangulations of X and Y respectively, or something along these lines. –  Andrey Dec 30 '13 at 1:19
    
Still cannot make any sense of the question. –  Alexandre Eremenko Dec 30 '13 at 1:47
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