An upper bound for the $k$-power domination number in $r$-uniform hypergraphs

Joseph S. Alameda, Franklin Kenter, Karen Meagher, Michael Young

Published 2020-04-16Version 1

The generalization of power domination, $k$-power domination, is a graph parameter denoted $\gamma_p^k$. The study of power domination was introduced to study electric power grids. Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. A new upper bound is also proven for $r\geq 3$.