Recently, optimization with graph cuts became very attractive but generally remains limited to small-scale problems due to the large memory requirement of graphs, even when restricted to binary variables. Unlike previous heuristics which generally fail to fully capture details, another band-based method was proposed for reducing these graphs in image segmentation. This method provides small graphs while preserving thin structures but do not offer low memory usage when the amount of regularization is large. This is typically the case when images are corrupted by an impulsive noise. In this paper, we overcome this situation by embedding a new parameter in this method to both further reducing graphs and filtering the segmentation. This parameter avoids any post-processing steps, appears to be generally less sensitive to noise variations and offers a good robustness against noise. We also provide an empirical way to automatically tune this parameter and illustrate its behavior for segmenting grayscale and color images.