Applications of deep learning to electronic design automation (EDA) have recently begun to emerge, although they have mainly been limited to processing of regular structured data such as images. However, many EDA problems require processing irregular structures, and it can be non-trivial to manually extract important features in such cases. In this paper, a high performance graph convolutional network (GCN) model is proposed for the purpose of processing irregular graph representations of logic circuits. A GCN classifier is firstly trained to predict observation point candidates in a netlist. The GCN classifier is then used as part of an iterative process to propose observation point insertion based on the classification results. Experimental results show the proposed GCN model has superior accuracy to classical machine learning models on difficult-to-observation nodes prediction. Compared with commercial testability analysis tools, the proposed observation point insertion flow achieves similar fault coverage with an 11% reduction in observation points and a 6% reduction in test pattern count.