Image segmentation is, in general, an ill-posed problem and additional constraints need to be imposed in order to achieve the desired result. Particularly in the field of medical image segmentation, a significant amount of prior knowledge is available that can be used to constrain the solution space of the segmentation problem. However, most of this prior knowledge is, in general, vague or imprecise in nature, which makes it very difficult to model. This is the problem that is addressed in this paper. Specifically, in this paper, we present fuzzy-cuts, a novel, knowledge-driven, graph-based method for medical image segmentation. We cast the problem of image segmentation as the maximum a posteriori (MAP) estimation of a Markov random field (MRF) which, in essence, is equivalent to the minimization of the corresponding Gibbs energy function. Considering the inherent imprecision that is common in the a priori description of objects in medical images, we propose a fuzzy theoretic model to incorporate knowledge-driven constraints into the MAP-MRF formulation. In particular, we focus on prior information about the object's location, appearance and spatial connectivity to a known seed region inside the object. To that end, we introduce fuzzy connectivity and fuzzy location priors that are used in combination to define the first-order clique potential of the Gibbs energy function. In our experiments, we demonstrate the application of the proposed method to the challenging problem of heart segmentation in non-contrast computed tomography (CT) data.