Despite recent advances, automatic blood vessel extraction from low quality retina images remains difficult. We propose an interactive approach that enables a user to efficiently obtain near perfect vessel segmentation with a few mouse clicks. Given two seed points, the approach seeks an optimal path between them by minimizing a cost function. In contrast to the Live-Vessel approach, the graph in our approach is based on the curve fragments generated with vessel tracing instead of individual pixels. This enables our approach to overcome the shortcut problem in extracting tortuous vessels and the problem of vessel interference in extracting neighboring vessels in minimal-cost path techniques, resulting in less user interaction for extracting thin and tortuous vessels from low contrast images. It also makes the approach much faster.