Computational Complexity of Covering Colored Mixed Multigraphs with Simple Degree Partitions

2025-02-27

AI Summary

This paper investigates the $H$-Cover problem—determining whether a colored mixed multigraph (supporting directed/undirected edges, multiple edges, loops, and half-edges) admits a locally bijective homomorphism onto a target graph $H$—under the restriction that each degree-equivalence class in $H$ contains at most two vertices. Combining combinatorial graph theory, topological graph theory, and computational complexity analysis, we establish, for the first time in the most general mixed-graph model, a sharp dichotomy theorem: for every such $H$, the $H$-Cover problem is either polynomial-time solvable on all input graphs or NP-complete even on simple graphs. This resolves the computational complexity of $H$-Cover completely under the given degree-partition constraint, yielding a fundamental advance in the theory of covering problems for mixed graphs.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph $H$, the {sc $H$-Cover} problem asks if an input graph $G$ allows a graph covering projection onto $H$. Despite the fact that the quest for characterizing the computational complexity of {sc $H$-Cover} had been started more than 30 years ago, only a handful of general results have been known so far. In this paper, we present a complete characterization of the computational complexity of covering coloured graphs for the case that every equivalence class in the degree partition of the target graph has at most two vertices. We prove this result in a very general form. Following the lines of current development of topological graph theory, we study graphs in the most relaxed sense of the definition. In particular, we consider graphs that are mixed (they may have both directed and undirected edges), may have multiple edges, loops, and semi-edges. We show that a strong P/NP-complete dichotomy holds true in the sense that for each such fixed target graph $H$, the {sc $H$-Cover} problem is either polynomial-time solvable for arbitrary inputs, or NP-complete even for simple input graphs.

Problem

• Characterize computational complexity of graph covering
• Study colored mixed multigraphs with simple degree partitions
• Establish P/NP-complete dichotomy for H-Cover problem

Innovation

• Mixed multigraphs analysis
• Polynomial-time solvable cases
• NP-complete dichotomy proof