Global properties, such as connectivity, shape constraints and boundary polarity, are useful high-level priors for image segmentation, allowing its customization for a given target object. In this work, we introduce a new method called Connected Oriented Image Foresting Transform (COIFT), which provides global optimum solutions according to a graph-cut measure, subject to the connectivity constraint in Oriented Image Foresting Transform (OIFT), ensuring the generation of connected objects, as well as allowing the simultaneous control of the boundary polarity. While the use of connectivity constraints in other frameworks, such as in the min-cut/max-flow algorithm, leads to an NP-Hard problem, COIFT conserves the low complexity of the OIFT algorithm. Experiments show that COIFT can considerably improve the segmentation of objects with thin and elongated parts, for the same number of seeds in segmentation based on markers.