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We present a fast graph cut algorithm for planar graphs. It is based on the graph theoretical work and leads to an efficient method that we apply on shape matching and image segmentation. In contrast to currently used methods in computer vision, the presented approach provides an upper bound for its runtime behavior that is almost linear. In particular, we are able to match two different planar shapes of N points in O(N2 log N) and segment a given image of N pixels in O(N log N). We present two experimental benchmark studies which demonstrate that the presented method is also in practice faster than previously proposed graph cut methods: On planar shape matching and image segmentation we observe a speed-up of an order of magnitude, depending on resolution.