Time series subsequence anomaly detection is an important task in a large variety of real-world applications ranging from health monitoring to AIOps, and is challenging due to the following reasons: 1) how to effectively learn complex dynamics and dependencies in time series; 2) diverse and complicated anomalous subsequences as well as the inherent variance and noise of normal patterns; 3) how to determine the proper subsequence length for effective detection, which is a required parameter for many existing algorithms. In this paper, we present a novel approach to subsequence anomaly detection, namely GraphSubDetector. First, it adaptively learns the appropriate subsequence length with a length selection mechanism that highlights the characteristics of both normal and anomalous patterns. Second, we propose a density-aware adaptive graph neural network (DAGNN), which can generate further robust representations against variance of normal data for anomaly detection by message passing between subsequences. The experimental results demonstrate the effectiveness of the proposed algorithm, which achieves superior performance on multiple time series anomaly benchmark datasets compared to state-of-the-art algorithms.