The Weighted Barycenter Drawing Recognition Problem

We consider the question of whether a given graph drawing $\mathrm{\Gamma }$ of a triconnected planar graph $G$ is a weighted barycenter drawing. We answer the question with an elegant arithmetic characterisation using the faces of $\mathrm{\Gamma }$. This leads to positive answers when the graph is a Halin graph, and to a polynomial time recognition algorithm when the graph is cubic.