ON INDEPENDENT AND (d, n)-DOMINATION NUMBERS OF HYPERCUBES

In this paper we consider the (d,n)-domination number, gamma(d,n) (Q(n)), the distance-d domination number gamma(d)(Q(n)) and the connected distance-d domination number gamma(c,d)(Q(n)) of n-dimensional hypercube graphs Q(n). We show that for 2 <= d <= [n/2], and n >= 4, gamma(d),(n)(Q(n)) <= 2(n-2d+2), improving the bound of Xie and Xu [19]. We also show that gamma(d)(Q(n)) <= 2(n-2d+2-r), for 2(r) - 1 <= n - 2d + 1 < 2(r+1) - 1, and gamma(c,d),(Q(n)) <= 2(n-d) for 1 <= n - d vertical bar 1 <= 3, and gamma(c,d)(Q(n)) <= 2(n -d-1) vertical bar 4, for n - d vertical bar 1 >= 4. Moreover, we give an upper bound of the independent domination number, gamma(i)(Q(n)) and the total domination number, gamma(t)(Q(n)) of Q(n). We show that gamma(i)(Q(n)) <= 2(n -k), gamma(t)(Q(n)) <= 2(n -k) for 2(k) - 1 < n < 2(k+1) - 1 and k >= 1 also we show that gamma(Q(n)) = gamma(i) (Q(n)) = 2(n -k) when n = 2(k) and k >= 3.