This paper presents a novel approach to visualize the uncertainty in graph-based segmentations of scalar data. Segmentation of 2D scalar data has wide application in a variety of scientific and medical domains. Typically, a segmentation is presented as a single unambiguous boundary although the solution is often uncertain due to noise or blur in the underlying data as well as imprecision in user input. Our approach provides insight into this uncertainty by computing the " min-path stability " , a scalar measure analyzing the stability of the segmentation given a set of input constraints. Our approach is efficient, easy to compute, and can be generally applied to either graph cuts or live-wire (even partial) segmentations. In addition to its general applicability, our new approach to graph cuts uncertainty visualization improves on the time complexity of the current state-of-the-art with an additional fast approximate solution. We also introduce a novel query enabled by our approach which provides users with alternate segmentations by efficiently extracting local minima of the segmentation optimization. Finally, we evaluate our approach and demonstrate its utility on data from scientific and medical applications.