$k$-tuple domination on Kneser graphs

Abstract

This paper considers multiple domination on Kneser graphs. We focus on $k$-tuple dominating sets, $2$-packings and the associated graph parameters $k$-tuple domination number and $2$-packing number. In particular, we compute the $2$-packing number of Kneser graphs $K(3r-2,r)$ and in odd graphs we obtain minimum $k$-tuple dominating sets of $K(7,3)$ and $K(11,5)$ for every $k$. Besides, we determine the Kneser graphs $K(n,r)$ with $k$-tuple domination number exactly $k+r$ and find all the minimum $k$-tuple dominating sets for these graphs, which generalize results for domination on Kneser graphs. Finally, we give a characterization of the $k$-tuple dominating sets of $K(n,2)$ in terms of the occurrences of the elements in $[n]$, which allows us to obtain minimum sized $k$-tuple dominating sets of $K(n,2)$ for $n\geq \Omega(\sqrt{k})$. Keywords: Kneser graphs, multiple domination, $k$-tuple domination, $2$-packings.