The notion of "strength of connectedness" between pixels has been successfully used in image segmentation. We present an extension to these works, which can considerably increase the efficiency of object definition tasks. A set of pixels is said a ?-connected component with respect to a seed pixel when the strength of connectedness of any pixel in that set with respect to the seed is higher than or equal to a threshold. While the previous approaches either assume no competition with a single threshold for all seeds or eliminate the threshold for seed competition, we found that seed competition with different thresholds can reduce the number of seeds and the need for user interaction during segmentation. We also propose automatic and user-friendly interactive methods for determining the thresholds. The improvements are demonstrated through several segmentation experiments involving medical images.