Grundy domination and zero forcing in Kneser graphs

BRESAR, BOSTJAN; KOS, TIM; TORRES, PABLO

ARS MATHEMATICA CONTEMPORANEA

Open Journal Systems

Año: 2019 vol. 17 p. 419 - 430

1855-3966

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 $ggrt(K_{n,r})$ equals ${{2r}choose{r}}$ for any $rge 2$ and $nge 2r+1$. For the Grundy domination number of Kneser graphs we get $ggr(K_{n,r})=alpha(K_{n,r})$ 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 $nge 3r+1$ and $rge 2$, while lower and upper bounds are provided for $Z(K_{n,r})$ when $2r+1le nle 3r$. Some lower bounds for different types of minimum ranks of Kneser graphs are also obtained along the way