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 [Kratochv\'{\ich}l: 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.

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