Graph Edit Distance (GED) is a widely used measure of graph similarity, valued for its flexibility in encoding domain knowledge through operation costs. However, existing learning-based approximation methods follow a modeling paradigm that decouples local candidate match selection from both operation costs and global dependencies between matches. This decoupling undermines their ability to capture the intrinsic flexibility of GED and often forces them to rely on costly iterative refinement to obtain accurate alignments. In this work, we revisit the formulation of GED and revise the prevailing paradigm, and propose Graph Edit Network (GEN), an implementation of the revised formulation that tightly integrates cost-aware expense estimation with globally guided one-step alignment. Specifically, GEN incorporates operation costs into node matching expenses estimation, ensuring match decisions respect the specified cost setting. Furthermore, GEN models match dependencies within and across graphs, capturing each match's impact on the overall alignment. These designs enable accurate GED approximation without iterative refinement. Extensive experiments on real-world and synthetic benchmarks demonstrate that GEN achieves up to a 37.8% reduction in GED predictive errors, while increasing inference throughput by up to 414x. These results highlight GEN's practical efficiency and the effectiveness of the revision. Beyond this implementation, our revision provides a principled framework for advancing learning-based GED approximation.

Subgraph matching is challenging as it necessitates time-consuming combinatorial searches. Recent Graph Neural Network (GNN)-based approaches address this issue by employing GNN encoders to extract graph information and hinge distance measures to ensure containment constraints in the embedding space. These methods significantly shorten the response time, making them promising solutions for subgraph retrieval. However, they suffer from scale differences between graph pairs during encoding, as they focus on feature counts but overlook the relative positions of features within node-rooted subtrees, leading to disturbed containment constraints and false predictions. Additionally, their hinge distance measures lack discriminative power for matched graph pairs, hindering ranking applications. We propose NC-Iso, a novel GNN architecture for neural subgraph matching. NC-Iso preserves the relative positions of features by building the hierarchical dependencies between adjacent echelons within node-rooted subtrees, ensuring matched graph pairs maintain consistent hierarchies while complying with containment constraints in feature counts. To enhance the ranking ability for matched pairs, we introduce a novel similarity dominance ratio-enhanced measure, which quantifies the dominance of similarity over dissimilarity between graph pairs. Empirical results on nine datasets validate the effectiveness, generalization ability, scalability, and transferability of NC-Iso while maintaining time efficiency, offering a more discriminative neural subgraph matching solution for subgraph retrieval.