Abstract
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r k (G,γ m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp.
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Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.
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Goddard, W., Henning, M.A. Restricted domination parameters in graphs. J Comb Optim 13, 353–363 (2007). https://doi.org/10.1007/s10878-006-9037-1
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DOI: https://doi.org/10.1007/s10878-006-9037-1