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?
PandQare triangulations ofXandYrespectively, or something along these lines. $\endgroup$