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

@inproceedings{Mane2012OnIA, title={On Independent and ( d , n )-domination numbers of hypercubes}, author={S. A. Mane and B. N. Waphare}, year={2012} }

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…

## 5 Citations

#A13 INTEGERS 21A (2021) SPANNING TREES AND DOMINATION IN HYPERCUBES

- Mathematics
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Let L(G) denote the maximum number of leaves in any spanning tree of a connected graph G. We show the (known) result that for the n-cube Qn, L(Qn) ∼ 2 = |V (Qn)| as n → ∞. Examining this more…

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Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow…

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- Computer Science, Mathematics
- 2022

We revisit the problem of determining the independent domination number in hypercubes for which the known upper bound is still not tight for general dimensions. We present here a constructive method…

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