We study approximation hardness of the MINIMUM DOMINATING SET problem and its variants in undirected and directed graphs. "We state the first explicit approximation lower bounds for varioiis kinds of domination problems (connected, total, independent) in bounded degree graphs. For most of dominating set problems we prove asymptotically almost tight lower bounds. The results are applied to improve the lower bounds for other related problems such as the MAXIMUM INDUCED MATCHING problem and the MAXIMUM LEAF SPANNING TREE problem.
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