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Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)
description
In 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.