$k$-domination invariants on Kneser graphsBrešar, Boštjan (Avtor) Dravec, Tanja (Avtor) Cornet, María Gracia (Avtor) Henning, Michael A. (Avtor) Kneser graphsk-dominationk-tuple total domination2-packingIn this follow-up to work of M.G. Cornet and P. Torres from 2023, where the $k$-tuple domination number and the $2$-packing number in Kneser graphs $K(n,r)$ were studied, we are concerned with two variations, the $k$-domination number, $\gamma_k(K(n,r))$, and the $k$-tuple total domination number, $\gamma_{t\times k}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that $\gamma_k(K(n,r))\ge \gamma_k(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and $\gamma_{t\times k}(K(n,r))\ge \gamma_{t\times k}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that $\gamma_k(K(n,r))= \gamma_{t\times k}(K(n,r))= k+r$ when $n\geq r(k+r)$, and that in this case every $\gamma_k$-set and $\gamma_{t\times k}$-set is a clique, while $\gamma_k(r(k+r)-1,r)=\gamma_{t\times k}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the $2$-packing number, $\rho_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $\rho_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $\rho_2(K(n,r))$ to be equal to some small values by imposing bounds on $r$ with respect to $n$. We also prove a version of monotonicity for the $2$-packing number of Kneser graphs. 20252025-07-24 13:34:10Neznano23096sl