Motion graphs have been widely successful in the synthesis of human motions. However, the quality of the generated motions depends heavily on the connectivity of the graphs and the quality of transitions in them. Achieving both of these criteria simultaneously though is difficult. Good connectivity requires transitions between less similar poses, while good motion quality requires transitions only between very similar poses. This paper introduces a new method for building motion graphs. The method first builds a set of interpolated motion clips, which contains many more similar poses than the original data set. The method then constructs a well-connected motion graph (wcMG), by using as little of the interpolated motion clip frames as necessary to provide good connectivity and only smooth transitions. Based on experiments, wcMGs outperform standard motion graphs across different measures, generate good quality motions, allow for high responsiveness in interactive control applications, and do not even require post-processing of the synthesized motions.