Grundy domination and zero forcing in Kneser graphs - Brešar, Boštjan (Ars mathematica contemporanea, 2019)
In this paper, we continue the investigation of different types of (Grundy) dominating sequences. We consider four different types of Grundy domination numbers and the related zero forcing numbers, focusing on these numbers in the well-known class of Kneser graphs ?$K_{n,r}$?. In particular, we establish that the Grundy total domination number ?$γ_{\rm gr}^t(K_{n,r})$? equals ?${{2r}\choose {r}}$? for any ?$r\ge 2$? and ?$n\ge 2r+1$?. For the Grundy domination number of Kneser graphs we get ?$γ_{\rm gr}(K_{n,r})=α(K_{n,r})$? whenever ?$n$? is sufficiently larger than ?$r$?. On the other hand, the zero forcing number ?$Z(K_{n,r})$? is proved to be ?${{n}\choose{r}}-{{2r}\choose{r}}$? when ?$n\ge 3r+1$? and ?$r\ge 2$?, while lower and upper bounds are provided for ?$Z(K_{n,r})$? when ?$2r+1\le n\le 3r$?. Some lower bounds for different types of minimum ranks of Kneser graphs are also obtained along the way.
Keywords: Grundy domination number; Grundy total domination number; Grundyjevo celotno dominatno število; Grundyjevo dominantno število; Kneser graph; Kneserjev graf; minimum rank; najmanjši rang; število ničelne prisile; zero forcing number