Graph Neural Networks (GNNs) have significantly advanced graph-level prediction tasks by utilizing efficient convolution and pooling techniques. However, traditional pooling methods in GNNs often fail to preserve key properties, leading to challenges such as graph disconnection, low decimation ratios, and substantial data loss. In this paper, we introduce three novel pooling methods based on Maximal Independent Sets (MIS) to address these issues. Additionally, we provide a theoretical and empirical study on the impact of these pooling methods on over-smoothing and over-squashing phenomena. Our experimental results not only confirm the effectiveness of using maximal independent sets to define pooling operations but also demonstrate their crucial role in mitigating over-smoothing and over-squashing.