Graph cuts have now become a cornerstone in computer vision for efficiently solving a numerous labeling problems. However, the large memory consumption of this method make it hard to solve large-scale problems. Except some exact methods, the heuristics present in the literature generally fail to fully capture shape complexities. In this paper, we first review the band-based strategy detailed in a previous paper for reducing graphs in image segmentation. Unlike exact techniques, the proposed method provide small graphs while accurately preserving thin structures. Next, an extra parameter is introduced for both further reducing graphs and removing small segments in the segmentation. While this parameter is generally less sensitive to variations, it offers a good robustness against noise. Finally, we provide an empirical way to automatically tune it and illustrate its behavior through experiments for segmenting grayscale and color images.