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Graph kernels and graph neural networks (GNNs) are widely used for the classification of graph data. However, many existing graph kernels and GNNs have limited expressive power, because they cannot distinguish graphs if the classic 1-dimensional Weisfeiler-Leman (1-WL) algorithm does not distinguish them. To break the 1-WL expressiveness barrier, we propose a novel method called filtration-enhanced graph transformation, which is based on a concept from the area of topological data analysis. In a nutshell, our approach first transforms each original graph into a filtration-enhanced graph based on a certain pre-defined filtration operation, and then uses the transformed graphs as the inputs for graph kernels or GNNs. The striking feature of our approach is that it is a plug-in method and can be applied in any graph kernel and GNN to enhance their expressive power. We theoretically and experimentally demonstrate that our solutions exhibit significantly better performance than the state-of-the art solutions for graph classification tasks.