Abstract
Given a graph G = (V, E) and an integer r ≥ 1, we call 'r-dominating code' any subset C of V such that every vertex in V is at distance at most r from at least one vertex in C. We investigate and locate in the complexity classes of the polynomial hierarchy, several problems linked with domination in graphs, such as, given r and G, the existence of, or search for, optimal r-dominating codes in G, or optimal r-dominating codes in G containing a subset of vertices X ⊂ V .
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