Pedestrian trajectory prediction is important in the research of mobile robot navigation in environments with pedestrians. Most pedestrian trajectory prediction algorithms require the input historical trajectories to be complete. If a pedestrian is unobservable in any frame in the past, then its historical trajectory become incomplete, the algorithm will not predict its future trajectory. To address this limitation, we propose the STGN-IT, a spatio-temporal graph network allowing incomplete trajectory input, which can predict the future trajectories of pedestrians with incomplete historical trajectories. STGN-IT uses the spatio-temporal graph with an additional encoding method to represent the historical trajectories and observation states of pedestrians. Moreover, STGN-IT introduces static obstacles in the environment that may affect the future trajectories as nodes to further improve the prediction accuracy. A clustering algorithm is also applied in the construction of spatio-temporal graphs. Experiments on public datasets show that STGN-IT outperforms state of the art algorithms on these metrics.