Erdős's 1985 growth question for critical edge sets in k-vertex-critical graphs is answered affirmatively for every k > 4 via f_k(n) = Ω(n^(1/3)), and a first upper bound f_k(n) = O(n/(log n)^Ω(1)) is given for all k ≥ 4.
VSR-Net: Vessel-like Structure Rehabilitation Network with Graph Clustering
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abstract
The morphologies of vessel-like structures, such as blood vessels and nerve fibres, play significant roles in disease diagnosis, e.g., Parkinson's disease. Deep network-based refinement segmentation methods have recently achieved promising vessel-like structure segmentation results. There are still two challenges: (1) existing methods have limitations in rehabilitating subsection ruptures in segmented vessel-like structures; (2) they are often overconfident in predicted segmentation results. To tackle these two challenges, this paper attempts to leverage the potential of spatial interconnection relationships among subsection ruptures from the structure rehabilitation perspective. Based on this, we propose a novel Vessel-like Structure Rehabilitation Network (VSR-Net) to rehabilitate subsection ruptures and improve the model calibration based on coarse vessel-like structure segmentation results. VSR-Net first constructs subsection rupture clusters with Curvilinear Clustering Module (CCM). Then, the well-designed Curvilinear Merging Module (CMM) is applied to rehabilitate the subsection ruptures to obtain the refined vessel-like structures. Extensive experiments on five 2D/3D medical image datasets show that VSR-Net significantly outperforms state-of-the-art (SOTA) refinement segmentation methods with lower calibration error. Additionally, we provide quantitative analysis to explain the morphological difference between the rehabilitation results of VSR-Net and ground truth (GT), which is smaller than SOTA methods and GT, demonstrating that our method better rehabilitates vessel-like structures by restoring subsection ruptures.
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Critical edge sets in vertex-critical graphs
Erdős's 1985 growth question for critical edge sets in k-vertex-critical graphs is answered affirmatively for every k > 4 via f_k(n) = Ω(n^(1/3)), and a first upper bound f_k(n) = O(n/(log n)^Ω(1)) is given for all k ≥ 4.