On the Complexity of Graph Edit Distance in Restricted Graph Classes

2026-07-21

AI Summary

This study investigates the computational complexity of graph edit distance restricted to specific graph classes such as paths and trees, examining how graph structure, vertex/edge labels, and edit cost functions influence its intractability. By establishing polynomial-time reductions from subgraph isomorphism and maximum common induced subgraph problems to graph edit distance, the work precisely delineates the complexity boundary between labeled and unlabeled settings: in the unlabeled case, the maximum common edge subgraph between paths and trees is solvable in polynomial time, whereas in the labeled case, the problem becomes NP-hard even when both input graphs are paths. Furthermore, the paper establishes a graph-class-preserving correspondence between maximum common edge subgraph and graph edit distance under specific cost functions, highlighting the pivotal role of labeling in determining the intrinsic difficulty of these problems.

Abstract

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.

Problem

• graph edit distance
• computational complexity
• restricted graph classes
• maximum common edge subgraph
• NP-hardness

Innovation

• graph edit distance
• computational complexity
• maximum common edge subgraph
• NP-hardness
• restricted graph classes