This article introduces a novel approach to generate visually promising skeletons automatically without any manual tuning. In practice, it is challenging to extract promising skeletons directly using existing approaches. This is because they either cannot fully preserve shape features, or require manual intervention, such as boundary smoothing and skeleton pruning, to justify the eye-level view assumption. We propose an approach here that generates backbone and dense skeletons by shape input, and then extends the backbone branches via skeleton grafting from the dense skeleton to ensure a well-integrated output. Based on our evaluation, the generated skeletons best depict the shapes at levels that are similar to human perception. To evaluate and fully express the properties of the extracted skeletons, we introduce two potential functions within the high-order matching protocol to improve the accuracy of skeleton-based matching. These two functions fuse the similarities between skeleton graphs and geometrical relations characterized by multiple skeleton endpoints. Experiments on three high-order matching protocols show that the proposed potential functions can effectively reduce the number of incorrect matches.