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This paper studies graph properties of the following forms: For every partition of the vertex set that satisfies an upper (or lower) bound on the number of elements in each partition class, there is a transversal of the partition that is an independent (or dominating) set. A possible application to fault-tolerant data storage is discussed, and bounds for the parameters that are functions of minimum and maximum degree are established. The complexity of associated decision problems is also addressed.
Michael R. Fellows (Tue,) studied this question.