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[...]

翻译：图覆盖概念是拓扑学中引入并深入研究的覆盖空间的离散化。在离散数学和理论计算机科学中，图覆盖从结构视角和复杂度视角都受到了广泛关注。然而，不连通图通常被排除在考量之外，理由是只需理解源图分量对目标图连通分量的覆盖即可。然而，连通图覆盖定义的不同（但等价）版本会推广为不连通图的非等价定义。本文旨在总结这一问题，并比较不连通图覆盖的三种不同方法：1）局部双射同态，2）全局满射的局部双射同态（我们称为“满射覆盖”），以及3）每个顶点被覆盖次数相同的局部双射同态（我们称为“均衡覆盖”）。我们比较的立足点是判定输入图是否覆盖给定目标图问题的复杂度。我们证明：满射覆盖和均衡覆盖均满足一个自然且受欢迎的性质——当且仅当对目标图的每个连通分量而言，判定问题可在多项式时间内解决时，覆盖不连通图的问题才是多项式时间可解的；若至少有一个分量是NP完全的，则覆盖问题也是NP完全的。我们进一步论证，第三种变体——均衡覆盖——是最自然的形式，尤其是在考虑染色图覆盖时。此外，从固定参数复杂度视角看，满射覆盖与均衡覆盖的复杂度存在差异。结合当前拓扑图论趋势及其在数学物理中的应用，我们以非常广义的方式考虑图...