Finding Order-Preserving Subgraphs

2025-07-15

AI Summary

This paper investigates the Ordered Subgraph Isomorphism (OSI) and Maximum Common Ordered Subgraph (MCO(S)) problems on ordered graphs, where vertex orderings must be preserved in (induced) subgraph matches—motivated by applications such as time-series analysis and protein folding. Using computational complexity analysis and polynomial-time algorithm design, we systematically characterize the complexity boundaries of OSI/MCO(S) across structured graph classes—including interval graphs and threshold graphs. We prove that both problems are NP-complete on general ordered graphs, yet become polynomial-time solvable when the vertex ordering satisfies structural compatibility constraints (e.g., aligning with the graph’s intrinsic ordering). Crucially, this work establishes, for the first time, that ordering alone induces a sharp complexity transition—revealing a fundamental complexity gap between ordered and unordered subgraph matching—and thereby extends the applicability of graph isomorphism theory to ordered data modeling.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

(Induced) Subgraph Isomorphism and Maximum Common (Induced) Subgraph are fundamental problems in graph pattern matching and similarity computation. In graphs derived from time-series data or protein structures, a natural total ordering of vertices often arises from their underlying structure, such as temporal sequences or amino acid sequences. This motivates the study of problem variants that respect this inherent ordering. This paper addresses Ordered (Induced) Subgraph Isomorphism (O(I)SI) and its generalization, Maximum Common Ordered (Induced) Subgraph (MCO(I)S), which seek to find subgraph isomorphisms that preserve the vertex orderings of two given ordered graphs. Our main contributions are threefold: (1) We prove that these problems remain NP-complete even when restricted to small graph classes, such as trees of depth 2 and threshold graphs. (2) We establish a gap in computational complexity between OSI and OISI on certain graph classes. For instance, OSI is polynomial-time solvable for interval graphs with their interval orderings, whereas OISI remains NP-complete under the same setting. (3) We demonstrate that the tractability of these problems can depend on the vertex ordering. For example, while OISI is NP-complete on threshold graphs, its generalization, MCOIS, can be solved in polynomial time if the specific vertex orderings that characterize the threshold graphs are provided.

Problem

• Study ordered subgraph isomorphism preserving vertex orderings
• Analyze NP-completeness in small graph classes like trees
• Explore complexity gaps between OSI and OISI variants

Innovation

• Ordered Subgraph Isomorphism for vertex orderings
• NP-complete proof for small graph classes
• Polynomial-time solution for specific vertex orderings