$k$-tuple domination on Kneser graphs

TL;DR

This paper investigates multiple domination parameters in Kneser graphs, providing exact values, characterizations, and minimum dominating sets for various cases, advancing understanding of domination properties in these combinatorial structures.

Contribution

It computes the 2-packing number for specific Kneser graphs, determines minimum k-tuple dominating sets for certain graphs, and characterizes k-tuple domination in K(n,2), generalizing previous results.

Findings

Computed 2-packing number of K(3r-2,r).

Found minimum k-tuple dominating sets for K(7,3) and K(11,5).

Characterized k-tuple dominating sets of K(n,2).

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.