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Graph cut is a popular technique for interactive image segmentation. However, it has certain shortcomings. In particular, graph cut has problems with segmenting thin elongated objects due to the ldquoshrinking biasrdquo. To overcome this problem, we propose to impose an additional connectivity prior, which is a very natural assumption about objects. We formulate several versions of the connectivity constraint and show that the corresponding optimization problems are all NP-hard. For some of these versions we propose two optimization algorithms: (i) a practical heuristic technique which we call DijkstraGC, and (ii) a slow method based on problem decomposition which provides a lower bound on the problem. We use the second technique to verify that for some practical examples DijkstraGC is able to find the global minimum.