Computational Complexity of Covering Colored Mixed Multigraphs with Simple Degree Partitions

TL;DR

This paper characterizes the computational complexity of the graph covering problem for a broad class of colored mixed graphs, establishing a clear P/NP-complete dichotomy based on the structure of the target graph.

Contribution

It provides a complete complexity classification for covering colored mixed graphs with degree partitions of size at most two, extending topological graph theory results.

Findings

The problem is either polynomial-time solvable or NP-complete for fixed target graphs.

A complete dichotomy is established based on the structure of the degree partition.

The results apply to very general graph models including mixed, multi, and semi-edges.

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.