Effective resistance in planar graphs and continued fractions

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{τ(G/e)}{τ(G)}$, where $τ(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees.