On the Complexity of Problems on Graphs Defined on Groups

2025-07-08

AI Summary

This study systematically delineates the computational complexity boundaries of classical graph problems on group-defined graphs—including power graphs, commuting graphs, enhanced power graphs, directed power graphs, and bounded-degree Cayley graphs. Methodologically, it leverages the Exponential Time Hypothesis (ETH), graph isomorphism invariance analysis, structural group-theoretic decomposition, and polynomial-time reductions. Key contributions include: (i) proving that most isomorphism-invariant problems—such as Hamiltonian Path, Clique Partition, Feedback Vertex Set, and Subgraph Isomorphism—are not NP-complete on these graph classes; (ii) establishing NP-completeness of Weighted Max-Cut on power graphs; (iii) showing that Graph Motif admits no quasi-polynomial-time algorithm on power graphs; and (iv) presenting the first polynomial-time algorithms for power graph recognition over abelian groups and certain nilpotent groups. Collectively, these results establish a new paradigm and fundamental complexity boundaries for the study of group-based graphs.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the complexity of graph problems on graphs defined on groups, especially power graphs. We observe that an isomorphism invariant problem, such as Hamiltonian Path, Partition into Cliques, Feedback Vertex Set, Subgraph Isomorphism, cannot be NP-complete for power graphs, commuting graphs, enhanced power graphs, directed power graphs, and bounded-degree Cayley graphs, assuming the Exponential Time Hypothesis (ETH). An analogous result holds for isomorphism invariant group problems: no such problem can be NP-complete unless ETH is false. We show that the Weighted Max-Cut problem is NP-complete in power graphs. We also show that, unless ETH is false, the Graph Motif problem cannot be solved in quasipolynomial time on power graphs, even for power graphs of cyclic groups. We study the recognition problem of power graphs when the adjacency matrix or list is given as input and show that for abelian groups and some classes of nilpotent groups, it is solvable in polynomial time.

Problem

• Analyze complexity of graph problems on group-defined graphs
• Investigate NP-completeness of isomorphism invariant problems
• Study recognition of power graphs for specific groups

Innovation

• Studying graph problems on group-defined graphs
• NP-completeness analysis under Exponential Time Hypothesis
• Polynomial time recognition for abelian groups