In this paper, we deal with the task of sub-event detection in evolving events using posts collected from the Twitter stream. By representing a sequence of successive tweets in a short time interval as a weighted graph-of-words, we are able to identify the key moments (sub-events) that compose an event using the concept of graph degeneracy. We then select a tweet to best describe each sub-event using a simple yet effective heuristic. We evaluated our approach using humangenerated summaries containing the actual important sub-events within each event and compare it to two baseline approaches using several performance metrics such as DET curves and precision/recall performance. Extensive experiments on recent sporting event streams indicate that our approach outperforms the dominant sub-event detection methods and constructs a humanreadable event summary by aggregating the most representative tweets of each sub-event.