For every rational resistance c/t, a simple planar graph with O(max(t/c, t/(t-c), log t)) vertices exists, and no graph can do better up to a constant.
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Effective resistance in planar graphs and continued fractions
For every rational resistance c/t, a simple planar graph with O(max(t/c, t/(t-c), log t)) vertices exists, and no graph can do better up to a constant.