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arXiv · 2205.06671

Improved Upper Bound on Independent Domination Number for Hypercubes

Debabani Chowdhury·Debesh K. Das·Bhargab B. Bhattacharya
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Abstract

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 to build an independent dominating set $S_n$ for the $n$-dimensional hypercube $Q_n$, where $n=2p+1$, $p$ being a positive integer $\ge 1$, provided an independent dominating set $S_p$ for the $p$-dimensional hypercube $Q_p$, is known. The procedure also computes the minimum independent dominating set for all $n=2^k-1$, $k>1$. Finally, we establish that the independent domination number $\alpha_n\leq 3 \times 2^{n-k-2}$ for $7\times 2^{k-2}-1\leq n<2^{k+1}-1$, $k>1$. This is an improved upper bound for this range as compared to earlier work.

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Debabani Chowdhury, Debesh K. Das, Bhargab B. Bhattacharya. 2022-05-13. Improved Upper Bound on Independent Domination Number for Hypercubes. https://arxiv.org/abs/2205.06671

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