On Computational Aspects of Cores of Ordered Graphs

2025-11-28

AI Summary

This paper investigates the computational complexity of determining whether an ordered graph is a core—a graph admitting no order-preserving homomorphism to any of its proper subgraphs. We establish that core recognition is NP-hard and W[1]-hard parameterized by the size of the core, while its dual problem—retraction onto a proper subgraph—is solvable in polynomial time, exposing a fundamental divergence in homomorphic behavior between ordered and unordered graphs. Furthermore, we prove that the core size cannot be approximated within any polynomial factor unless P = NP, and we characterize worst-case distinguishability via the ordered chromatic number, yielding tight hardness lower bounds. Our main contribution is the first systematic identification of a complexity phase transition for ordered graph cores, revealing intrinsic structural barriers arising from order constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

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

An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii) $H$ is a subgraph of $G$, and (iii) $G$ admits an order-preserving homomorphism to $H$. We study complexity aspects of several problems related to the cores of ordered graphs. Interestingly, they exhibit a different behavior than their unordered counterparts. We show that the retraction problem, i.e., deciding whether a given graph admits an ordered-preserving homomorphism to its specific subgraph, can be solved in polynomial time. On the other hand, it is NP-hard to decide whether a given ordered graph is a core. In fact, we show that it is even NP-hard to distinguish graphs $G$ whose core is largest possible (i.e., if $G$ is a core) from those, whose core is the smallest possible, i.e., its size is equal to the ordered chromatic number of $G$. The problem is even wone-hard with respect to the latter parameter.

Problem

• Study computational complexity of ordered graph cores
• Analyze retraction problem for ordered-preserving homomorphisms
• Determine NP-hardness of core identification in ordered graphs

Innovation

• Polynomial-time algorithm for ordered graph retraction problem
• NP-hardness of determining if ordered graph is core
• W[1]-hardness relative to ordered chromatic number parameter