On graphs maximizing the zero forcing number.

A subset Z of vertex set V (G) of a graph G is a zero forcing set of G if iteratively adding vertices to Z from V (G) ∖ Z that are the unique neighbor in V (G) ∖ Z of some vertex in Z , results in the entire V (G) of G. The zero forcing number Z (G) of G is the minimum cardinality of a zero forcing set of G. In a recent work, Caro and Pepper (2015) proved that Z (G) ≤ (Δ − 2) n − (Δ − δ) + 2 Δ − 1 , where n , Δ , δ are the order, maximum degree, minimum degree of G , respectively. In this paper we completely characterize extremal graphs attained the upper bound, solving a problem proposed by Lu et al. (2016). Our result further generalizes the characterization of extremal regular graphs obtained by Gentner et al. and Lu et al. in 2016. [ABSTRACT FROM AUTHOR]