Grundy domination and zero forcing in Kneser graphs
Boštjan Brešar, Tim Kos, Pablo Torres · 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 gr t ( K n , r ) equals (2 r choose r ) for any r ≥ 2 and n ≥ 2 r + 1 . For the Grundy domination number of Kneser graphs we get γ 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 ) − (2 r choose r ) when n ≥ 3 r + 1 and r ≥ 2 , while lower and upper bounds are provided for Z ( K n , r ) when 2 r + 1 ≤ n ≤ 3 r . Some lower bounds for different types of minimum ranks of Kneser graphs are also obtained along the way.