Graph covers and semi-covers: Who is stronger?
Jan Kratochvil, Roman Nedela
TL;DR
The paper investigates when a graph $A$ is stronger than a graph $B$ in the sense of graph coverings, formalizing the relation $A▷B$ and its connection to the NP-hardness dichotomy for H-Cover. It introduces semi-covers as a key tool and develops product-like constructions, notably $G^×$ and $G^⊙$, to analyze and construct covers. The main results show that for the one-vertex cubic targets $F(3,0)$ and $F(1,1)$, the stronger relation coincides with the usual covering relation in broad cases: $A▷F(3,0)$ iff $A→F(3,0)$ for all $A$, and $A▷F(1,1)$ iff $A\leadsto F(1,1)$, with a complete cubic-classification for graphs with up to four vertices. These findings advance understanding of the strong-dichotomy phenomenon in restricted cubic settings and provide a framework (via semi-covers) for analyzing stronger-than relations with potential implications for complexity reductions.
Abstract
The notion of graph cover, also known as locally bijective homomorphism, is a discretization of covering spaces known from general topology. It is a pair of incidence-preserving vertex- and edge-mappings between two graphs, the edge-component being bijective on the edge-neighborhoods of every vertex and its image. In line with the current trends in topological graph theory and its applications in mathematical physics, graphs are considered in the most relaxed form and as such they may contain multiple edges, loops and semi-edges. Nevertheless, simple graphs (binary structures without multiple edges, loops, or semi-edges) play an important role. It has been conjectured in [Bok et al.: List covering of regular multigraphs, Proceedings IWOCA 2022, LNCS 13270, pp. 228--242] that for every fixed graph $H$, deciding if a graph covers $H$ is either polynomial time solvable for arbitrary input graphs, or NP-complete for simple ones. A graph $A$ is called stronger than a graph $B$ if every simple graph that covers $A$ also covers $B$. This notion was defined and found useful for NP-hardness reductions for disconnected graphs in [Bok et al.: Computational complexity of covering disconnected multigraphs, Proceedings FCT 2022, LNCS 12867, pp. 85--99]. It was conjectured in [Kratochvíl: Towards strong dichotomy of graphs covers, GROW 2022 - Book of open problems, p. 10, {\tt https://grow.famnit.upr.si/GROW-BOP.pdf}] that if $A$ has no semi-edges, then $A$ is stronger than $B$ if and only if $A$ covers $B$. We prove this conjecture for cubic one-vertex graphs, and we also justify it for all cubic graphs $A$ with at most 4 vertices.