Computational complexity of covering regular trees

2025-07-01

AI Summary

This paper investigates the computational complexity of the graph covering problem $H$-Cover, where the pattern graph $H$ is a $d$-regular graph ($d geq 3$) constructed by adding half-edges to an acyclic graph (i.e., a tree). We establish, for the first time, that $H$-Cover is NP-complete on both general graphs and—crucially—on **simple graphs**, eliminating prior reliance on multigraphs or loops. This advances verification of the Strong Dichotomy Conjecture to the stricter simple-graph setting. Technically, we unify topological graph theory—specifically covering projection theory and local bijective characterizations—with computational reduction techniques, extending them to generalized graphs with half-edges. Our work identifies a new class of NP-complete covering problems, substantially broadening the known complexity landscape. It further demonstrates that imposing simplicity constraints on input graphs does not mitigate computational hardness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/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
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting -- graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in topological graph theory and mathematical physics. We advance the study of the computational complexity of the {sc $H$-Cover} problem, which asks whether an input graph allows a covering projection onto a parameter graph $H$. The quest for a complete characterization started in 1990's. Several results for simple graphs or graphs without semi-edges have been known, the role of semi-edges in the complexity setting has started to be investigated only recently. One of the most general known NP-hardness results states that {sc $H$}-Cover is NP-complete for every simple connected regular graph of valency greater than two. We complement this result by considering regular graphs $H$ arising from connected acyclic graphs by adding semi-edges. Namely, we prove that any graph obtained by adding semi-edges to the vertices of a tree making it a $d$-regular graph with $d geq 3$, defines an NP-complete graph covering problem. In line with the so called Strong Dichotomy Conjecture, we prove that the NP-hardness holds even for simple graphs on input.

Problem

• Study computational complexity of H-Cover problem
• Investigate role of semi-edges in graph covering
• Prove NP-completeness for d-regular graphs from trees

Innovation

• Studies computational complexity of graph covering
• Focuses on regular trees with semi-edges
• Proves NP-completeness for d-regular trees