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Let G be a graph with degree sequence d 1 ≥ ⋯ ≥ d n . Slater proposed s ℓ ( G ) = min { s : ( d 1 + 1 ) + ⋯ + ( d s + 1 ) ≥ n } as a lower bound on the domination number γ ( G ) of G . We show that deciding the equality of γ ( G ) and s ℓ ( G ) for a given graph G is NP-complete but that one can decide efficiently whether γ ( G ) > s ℓ (...
As a natural variant of domination in graphs, Dankelmann et al. (2009) introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if S is a set of vertices of a graph G , then S...
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