On the Complexity of Graph Edit Distance in Restricted Graph Classes

The graph edit distance generalizes several well-known NP-hard problems and is therefore NP-hard itself. However, the relationship between the considered graph class, the edit cost function, and the resulting computational complexity is not well understood. We investigate this interplay by revisiting polynomial-time reductions from the literature, which reduce subgraph isomorphism and maximum common induced subgraph to the graph edit distance. For these classical problems, a sharp distinction between NP-hard and polynomial-time solvable cases is known, and we make the implications for the complexity of the graph edit distance explicit. We establish a graph-class-preserving correspondence between the maximum common edge subgraph and graph edit distance under a specific cost function, both in labeled and unlabeled graphs. In the unlabeled setting, the maximum common edge subgraph problem is polynomial-time solvable when one graph is a path and the other is a tree. In contrast, for labeled graphs, we prove that both the maximum common edge subgraph and the graph edit distance remain NP-hard, even when both graphs are paths.