Hardness of r-dominating set on Graphs of Diameter (r + 1)
The dominating set problem has been extensively studied in the realm of parameterized complexity. It is one of the most common sources of reductions while proving the parameterized intractability of problems. In this paper, we look at dominating set and its generalization r-dominating set on graphs of bounded diameter in the realm of parameterized complexity. We show that dominating set remains W-hard on graphs of diameter 2, while r-dominating set remains W-hard on graphs of diameter r + 1. The lower bound on the diameter in our intractability results is the best possible, as r-dominating set is clearly polynomial time solvable on graphs of diameter at most r.
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