Tightening Curves on Surfaces Monotonically with Applications

We prove the first polynomial bound on number of monotonic homotopy moves required to tighten a collection closed curves any compact orientable surface, where crossings in curve is not allowed increase at time during process. The best known upper before was exponential, which can be obtained by combining algorithm De Graaf and Schrijver [ J. Comb. Theory Ser. B , 1997] together with an exponential possible surface maps. To obtain new bound, we apply tools from hyperbolic geometry, as well operations graph drawing algorithms—the cluster pipe expansions—to study surfaces. As corollaries, present two efficient algorithms for graphs First, provide polynomial-time convert given multicurve into minimal position. Such only existed single curves, it that previous techniques do generalize case. Second, reduce k -terminal plane (and more generally, graph) using degree-1 reductions, series-parallel Δ Y -transformations arbitrary integer . Previous planar setting when ≤ 4, all them rely extensive case-by-case analysis based different values Our makes use connection between electrical transformations thus solves problem unified fashion.