Lower Bounds for Electrical Reduction on Surfaces

We strengthen the connections between electrical transformations and homotopy from the planar setting---observed and studied since Steinitz---to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the number of electrical transformations required to reduce an $n$-vertex graph on surface in the worst case [SOCG 2016] in two different directions. Our previous $Ω(n^{3/2})$ lower bound applies only to facial electrical transformations on plane graphs with no terminals. First we provide a stronger $Ω(n^2)$ lower bound when the planar graph has two or more terminals, which follows from a quadratic lower bound on the number of homotopy moves in the annulus. Our second result extends our earlier $Ω(n^{3/2})$ lower bound to the wider class of planar electrical transformations, which preserve the planarity of the graph but may delete cycles that are not faces of the given embedding. This new lower bound follows from the observation that the defect of the medial graph of a planar graph is the same for all its planar embeddings.