  The notion of graph covers is a discretization of covering spaces introduced
and deeply studied in topology. In discrete mathematics and theoretical
computer science, they have attained a lot of attention from both the
structural and complexity perspectives. Nonetheless, disconnected graphs were
usually omitted from the considerations with the explanation that it is
sufficient to understand coverings of the connected components of the target
graph by components of the source one. However, different (but equivalent)
versions of the definition of covers of connected graphs generalize to
non-equivalent definitions for disconnected graphs. The aim of this paper is to
summarize this issue and to compare three different approaches to covers of
disconnected graphs: 1) locally bijective homomorphisms, 2) globally surjective
locally bijective homomorphisms (which we call \emph{surjective covers}), and
3) locally bijective homomorphisms which cover every vertex the same number of
times (which we call \emph{equitable covers}). The standpoint of our comparison
is the complexity of deciding if an input graph covers a fixed target graph. We
show that both surjective and equitable covers satisfy what certainly is a
natural and welcome property: covering a disconnected graph is polynomial-time
decidable if such it is for every connected component of the graph, and it is
NP-complete if it is NP-complete for at least one of its components. We further
argue that the third variant, equitable covers, is the most natural one, namely
when considering covers of colored graphs. Moreover, the complexity of
surjective and equitable covers differ from the fixed parameter complexity
point of view.
  In line with the current trends in topological graph theory, as well as its
applications in mathematical physics, we consider graphs in a very general
sense[...]
