Restricted domination parameters in graphs

13 years 4 months ago

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (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 γm, and of an m-tuple dominating set is γ×m. For a property π of subsets of V (G), with associated parameter fπ, the k-restricted π-number rk(G, fπ) 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 rk(G, γm) ≤ (mn+k)/(m+1); (b) if G has minimum degree 3, then rk(G, γ) ≤ (3n + 5k)/8; and (c) if G is connected with minimum degree at least 2, then rk(G, γ×2) ≤ 3n/4 + 2k/7. These bounds are sharp.

Wayne Goddard, Michael A. Henning

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