Computational Complexity of Covering Colored Mixed Multigraphs with Simple Degree Partitions

TL;DR

This work provides a complete complexity dichotomy for the H-Cover problem on colored mixed multigraphs when every degree-partition class has at most two vertices. It develops a general framework around covering projections, degree partitions, and block graphs, and proves that the problem is polynomial-time solvable in the tractable regime via a novel 2-SAT plus matching algorithm, while NP-completeness arises in the presence of harmful or dangerous monochromatic block graphs, even for simple inputs. The NP-hard results are established through intricate gadget constructions (e.g., limping tripod) and reductions from 2-in-4-SAT, complemented by a garbage-collection technique to assemble partial coverings into full projections. A key exception is FW(2), which remains polynomial-time solvable, though it can trigger NP-hardness under degree constraints. Overall, the paper advances the understanding of the graph-cover complexity landscape and offers a robust method that blends logical and combinatorial tools to classify tractability for a broad class of target graphs.

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.

Figures (6)

• Figure 1: An example of a covering projection from a graph $G$ to a graph $H$.
• Figure 2: An example of the application of the degree adjusting reduction.
• Figure 3: Examples of the considered small graphs.
• Figure 4: The maximal harmless monochromatic uniblock (left) and interblock (right) graphs ($c$ is an arbitrary non-negative integer).
• Figure 6: Block graphs forced by $FW(2)$.

Theorems & Definitions (20)

• Conjecture : Strong Dichotomy Conjecture for Graph Covers n:BFJKR24-Algorithmica
• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 5
• Definition 6
• Definition 8
• Theorem 9
• Lemma 10