On Independent and (d, n)-domination numbers of hypercubes

In this paper we consider the (d, n)-domination number, γd,n(Qn), the distance-d domination number γd(Qn) and the connected distance-d domination number γc,d(Qn) of ndimensional hypercube graphs Qn. We show that for 2 ≤ d ≤ bn/2c, and n ≥ 4, γd,n(Qn) ≤ 2n−2d+2, improving the bound of Xie and Xu [19]. We also show that γd(Qn) ≤ 2n−2d+2−r, for 2 − 1 ≤ n − 2d + 1 < 2 − 1, and γc,d(Qn) ≤ 2n−d, for 1 ≤ n− d+ 1 ≤ 3, and γc,d(Qn) ≤ 2n−d−1 + 4, for n− d+ 1 ≥ 4. Moreover, we give an upper bound of the independent domination number, γi(Qn) and the total domination number, γt(Qn) of Qn. We show that γi(Qn) ≤ 2n−k, γt(Qn) ≤ 2n−k for 2 − 1 < n < 2− 1 and k ≥ 1 also we show that γ(Qn) = γi(Qn) = 2n−k when n = 2 and k ≥ 3.