Complexity Aspects of Homomorphisms of Ordered Graphs

2025-11-28

AI Summary

This paper investigates the computational and parameterized complexity of homomorphisms on ordered graphs—graphs whose vertices are equipped with a linear order. Methodologically, it employs reduction techniques—including an embedding of unordered graphs into ordered bipartite graphs—to establish NP-completeness and to show that the problem lies in XP but is W[1]-hard when parameterized by the number of vertices in the target graph. The main contribution is the first systematic complexity classification framework for ordered graph homomorphisms, identifying key tractable subclasses—such as ordered interval graphs and ordered convex graphs—for which the problem is solvable in polynomial time. The paper further designs efficient algorithms for these classes and reveals the fundamental role of vertex ordering: while it can induce complexity jumps (e.g., from P to NP-hard), it also enables novel algorithmic approaches unavailable in the unordered setting.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along with algorithms associated with related problems. These questions are interesting, and we show that numerous problems lead to various complexities. The reduction from homomorphisms of unordered structures to homomorphisms of ordered graphs is proved, achieved with the use of ordered bipartite graphs. We then determine the NP-completeness of the problem of finding ordered homomorphisms of ordered graphs and the XP and W[1]-hard nature of this problem parameterized by the number of vertices of the image ordered graph. Classes of ordered graphs for which this problem can be solved in polynomial time are also presented.

Problem

• Analyzing computational complexity of homomorphisms in ordered graphs
• Proving NP-completeness of finding ordered graph homomorphisms
• Identifying polynomial-time solvable classes for ordered homomorphisms

Innovation

• Reduction via ordered bipartite graphs
• NP-completeness of ordered homomorphisms
• Polynomial-time classes for ordered graphs