05-XX COMBINATORICS (For finite fields, see 11Txx)
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While graph covering is a fundamental and well-studied problem, this field lacks a broad and unified literature review. The holistic overview of graph covering given in this article attempts to close this gap. The focus lies on a characterization and classification of the different problems discussed in the literature. In addition, notable results and common approaches are also included. Whenever appropriate, our review extends to the corresponding partioning problems.
We consider the problem of partitioning a weighted graph into k
connected components of similar weight. In particular, we consider the two classical objectives to maximize the lightest part or to minimize the heaviest part. For a partitioning of the vertex set and for both objectives, we give the first known approximation results on general graphs. Specifically, we give a $\Delta$-approximation where $\Delta$ is the maximum degree of an arbitrary spanning tree of the given graph.
Concerning the edge partition case, we even obtain a 2-approximation for the min-max and the max-min problem, by using the claw-freeness of line graphs.