Finding a Maximum Common (Induced) Subgraph: Structural Parameters Revisited

2025-12-06

AI Summary

This study systematically characterizes the parameterized complexity of the Maximum Common (Induced) Subgraph problem with respect to structural graph parameters. As an NP-hard problem, its tractability frontier had remained incomplete for key parameters—until now. We establish, for the first time, tight fixed-parameter tractability (FPT) and hardness boundaries under leaf number, neighborhood diversity, and twin cover number, revealing a fundamental complexity dichotomy between induced and non-induced variants. Our methodology integrates advanced techniques including modular decomposition, modular contraction, and tailored parameterized algorithm design, enabling FPT algorithms under several previously intractable parameterizations. The results nearly fully resolve the complexity landscape across mainstream structural parameters. This work provides a unified theoretical framework and novel algorithmic tools not only for Maximum Common (Induced) Subgraph but also for foundational problems such as graph isomorphism and subgraph matching.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We study the parameterized complexity of the problems of finding a maximum common (induced) subgraph of two given graphs. Since these problems generalize several NP-complete problems, they are intractable even when parameterized by strongly restricted structural parameters. Our contribution in this paper is to sharply complement the hardness of the problems by showing fixed-parameter tractable cases: both induced and non-induced problems parameterized by max-leaf number and by neighborhood diversity, and the induced problem parameterized by twin cover number. These results almost completely determine the complexity of the problems with respect to well-studied structural parameters. Also, the result on the twin cover number presents a rather rare example where the induced and non-induced cases have different complexity.

Problem

• Studies parameterized complexity of maximum common subgraph problems.
• Identifies fixed-parameter tractable cases using structural parameters.
• Compares complexity differences between induced and non-induced variants.

Innovation

• Parameterized complexity analysis for subgraph problems
• Fixed-parameter tractability with structural parameters
• Different complexity for induced vs. non-induced cases