Minimum cost paths have been extensively studied theoretical tools for interactive image segmentation. The existing geodesically linked active contour (GLAC) model, which basically consists of a set of vertices connected by paths of minimal cost, blends the benefits of minimal paths and region-based active contours. This results in a closed piecewise-smooth curve, over which an edge or region energy functional can be formulated. As an important shortcoming, the GLAC in its initial formulation does not guarantee the curve to be simple, consistent with respect to the purpose of segmentation. In this paper, we draw our inspiration from the GLAC and other boundary-based interactive segmentation algorithms, in the sense that we aim to extract a contour given a set of user-provided points, by connecting these points using paths. The key idea is to select a combination among a set of possible paths, such that the resulting structure represents a relevant closed curve. Instead of considering minimal paths only, we switch to a more general formulation, which we refer to as admissible paths. These basically correspond to the roads travelling along the bottom of distinct valleys between given endpoints. We introduce a novel term to favor the simplicity of the generated contour, as well as a local search method to choose the best combination among possible paths.