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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, providing a linear algebra approach to the domination problem. We show that several Grundy domination type parameters are bounded above by 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. [ABSTRACT FROM AUTHOR]

The maximum nullity M ( G ) and the Colin de Verdière type parameter ξ ( G ) both consider the largest possible nullity over matrices in S ( G ) , which is the family of real symmetric matrices whose i , j -entry, i ≠ j , is nonzero if i is adjacent to j , and zero otherwise; however, ξ ( G ) restricts to those matrices A in S ( G ) with the Strong Arnold Property, which means X = O is the only symmetric matrix that satisfies A ∘ X = O , I ∘ X = O , and A X = O . This paper introduces zero forcing parameters Z SAP ( G ) and Z vc ( G ) , and proves that Z SAP ( G ) = 0 implies every matrix A ∈ S ( G ) has the Strong Arnold Property and that the inequality M ( G ) − Z vc ( G ) ≤ ξ ( G ) holds for every graph G . Finally, the values of ξ ( G ) are computed for all graphs up to 7 vertices, establishing ξ ( G ) = ⌊ Z ⌋ ( G ) for these graphs. [ABSTRACT FROM AUTHOR]