The maximum mean cycle weight (MMCW) segmentation framework is a graph-based alternative to approaches such as GraphCut or Markov Random Fields. It offers time- and space-efficient computation and guaranteed optimality. However, unlike GraphCut or Markov Random Fields, MMCW does not seek to segment the entire image, but rather to find the single best object within the image, according to an objective function encoded by edge weights. Its focus on a single, best object makes MMCW attractive to interactive segmentation settings, where the user indicates which objects are to be segmented. However, a provably correct way of performing interactive segmentation using the MMCW framework has never been established. Further, the question of how to develop a good objective function based on user-provided information has never been addressed. Here, we propose a three-component objective function specifically designed for use with interactive MMCW segmentation. Two of those components, representing object boundary and object interior information, can be learned from a modest amount of user-labelled data, but in a way unique to the MMCW framework. The third component allows us to extend the MMCW framework to the situation of interactive segmentation. Specifically, we show that an appropriate weighted combination of the three components guarantees that the object produced by MMCW segmentation will enclose user-specified pixels that can be chosen interactively. The component weights can either be computed a priori based on image characteristics, or online via an adaptive reweighting scheme. We demonstrate the success of the approach on several microscope image segmentation problems.