Accurately restoring topology is both challenging and crucial in tubular structure extraction tasks, such as blood vessel segmentation and road network extraction. Diverging from traditional approaches based on pixel-level classification, our proposed method, named GraphMorph, focuses on branch-level features of tubular structures to achieve more topologically accurate predictions. GraphMorph comprises two main components: a Graph Decoder and a Morph Module. Utilizing multi-scale features extracted from an image patch by the segmentation network, the Graph Decoder facilitates the learning of branch-level features and generates a graph that accurately represents the tubular structure in this patch. The Morph Module processes two primary inputs: the graph and the centerline probability map, provided by the Graph Decoder and the segmentation network, respectively. Employing a novel SkeletonDijkstra algorithm, the Morph Module produces a centerline mask that aligns with the predicted graph. Furthermore, we observe that employing centerline masks predicted by GraphMorph significantly reduces false positives in the segmentation task, which is achieved by a simple yet effective post-processing strategy. The efficacy of our method in the centerline extraction and segmentation tasks has been substantiated through experimental evaluations across various datasets. Source code will be released soon.