Zero forcing number, Grundy domination number, and their variants

Abstract

This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem. We show that the Grundy domination type parameters are bounded above by the minimum rank type parameters. We also give a method to calculate the L-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the L-Grundy domination number.

Grundy domination and zero forcing in Kneser graphs

“…In addition, the skew zero forcing number (denoted by Z − (G)) was introduced in [18], and studied in the context of the invariant mr 0 , which is a version of the minimum rank in which matrices are in addition required to have empty diagonals. Motivated by the results of [5], Lin [20] noticed a similar connection between the Grundy total domination number and the skew zero forcing number of graphs, and also between the Grundy domination number and another version of the minimum rank parameter, denoted by mr˙ . As shown by Lin [20], the following bounds hold:…”

“…Motivated by the results of [5], Lin [20] noticed a similar connection between the Grundy total domination number and the skew zero forcing number of graphs, and also between the Grundy domination number and another version of the minimum rank parameter, denoted by mr˙ . As shown by Lin [20], the following bounds hold:…”

“…As mentioned earlier, Lin established the connections between different Grundy domination numbers, zero forcing numbers and minimum rank invariants. (For definitions of different zero forcing and minimum rank parameters we refer to [20].) 20]).…”

On Grundy total domination number in product graphs

“…The corresponding invariants obtained from the longest possible (Z-or L-) sequences are denoted by γ Z gr (G) and γ L gr (G), respectively. As it turns out, these two invariants have natural counterparts also in the zero-forcing and minimum-rank world, see [17].…”

“…(The zero forcing number is in turn very useful in determining the minimum rank of a graph [2].) Lin [17] noticed a similar connection between the Grundy total domination number and the skew zero forcing number of graphs, where the skew zero forcing number was introduced in [15] and is denoted by Z − (G). (For some recent related results see [1,3,4,5,12,16,18].)…”