In this work, a method based on optimum cuts in graphs is proposed for unsupervised image segmentation, that can be tailored to different objects, according to their boundary polarity, by extending the Oriented Image Foresting Transform (OIFT). The proposed method, named UOIFT, encompasses as a particular case the single-linkage algorithm by minimum spanning tree (MST), establishing important theoretical contributions, and gives superior segmentation results compared to other approaches commonly used in the literature, usually requiring a lower number of image partitions to isolate the desired regions of interest. The method is supported by new theoretical results involving the usage of non-monotonic-incremental cost functions in directed graphs. The results are demonstrated using a region adjacency graph of superpixels in medical and natural images.