On Double Domination in Graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). A function f(p) is defined, and it is shown that γ×2(G) = min f(p), where the minimum is taken over the n-dimensional cube C = {p = (p1, . . . , pn) | pi ∈ IR, 0 ≤ pi ≤ 1, i = 1, . . . , n}. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G) ≤ ((ln(1 + d) + ln δ + 1)/δ)n.