Abstract: Subgraph matching is a fundamental building block for graph-based applications and is challenging due to its high-order combinatorial nature. Existing studies usually tackle it by combinatorial optimization or learning-based methods. However, they suffer from exponential computational costs or searching the matching without theoretical guarantees. In this paper, we develop $D^2$Match by leveraging the efficiency of Deep learning and Degeneracy for subgraph matching. More specifically, we first prove that subgraph matching can degenerate to subtree matching, and subsequently is equivalent to finding a perfect matching on a bipartite graph. We can then yield an implementation of linear time complexity by the built-in tree-structured aggregation mechanism on graph neural networks. Moreover, circle structures and node attributes can be easily incorporated in $D^2$Match to boost the matching performance. Finally, we conduct extensive experiments to show the superior performance of our $D^2$Match and confirm that our $D^2$Match indeed exploits the subtrees and differs from existing GNNs-based subgraph matching methods that depend on memorizing the data distribution divergence.