Let P be a complex polynomial of degree \(n\ge 2\) in the Riemann sphere, and let \(\gamma \) be an oriented Jordan path running through its critical values. A classical algorithm, rooted in the pioneering work of H. A. Schwarz and F. Klein, states that the inverse image of \(\gamma \) under P determines a finite tessellation of the Riemann sphere, with tiles that are topological k–polygons and alternate colors. Following a question by W. P. Thurston, we study under what conditions a finite graph (or equivalently a finite tessellation of the Riemann sphere) originates from a generic polynomial P and an oriented Jordan path \(\gamma \) as above.
