Grain size plays a fundamental role in the mechanical properties of materials. Recently, automatic measurement of average grain size attacks more and more attention based on computer vision. However, low contrast, twin grains, thin boundary, and low connectivity limit the achievement of automatic and accurate image analysis. Inspired by the calculation procedure of grain size, we propose a center-guided and connectivity-preserving network for grain boundaries segmentation. On one hand, the proposed center feature extraction module and center-guided feature recalibration mechanism (CFRM) make the network pay more attention to the center area. On the other hand, a connectivity-preserving loss function is integrated with the network, which forces the network to converge toward high connectivity. Benefiting from the above aspects, our network can segment the grain boundary with high structural integrity and avoid the complex post-processing process. Experiments on the SRIF-GSM dataset reveal that our method achieves 85.98 mIoU and 95.60 clDice scores, demonstrating significant advantages compared with the state-of-the-art semantic segmentation methods.