Improved Upper Bound on Independent Domination Number for Hypercubes

TL;DR

This paper introduces a new constructive method to find independent dominating sets in hypercubes, improving the upper bounds on their independent domination number for certain dimensions.

Contribution

It presents a recursive construction for independent dominating sets in hypercubes and establishes a tighter upper bound for the independent domination number in specific dimension ranges.

Findings

Constructive method for hypercube independent dominating sets

Improved upper bound on independent domination number

Applicable to dimensions of the form 2^k - 1

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.