Extremal values and bounds for the zero forcing number

A set Z of vertices of a graph G is a zero forcing set of G if iteratively adding to Z vertices from V (G) \ Z that are the unique neighbor in V (G) \ Z of some vertex in Z, results in the entire vertex set V(G) of G. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. Amos et al. (2015) proved Z(G) <= ((Delta - 2)n + 2)/(Delta - 1) for a connected graph G of order n and maximum degree Delta >= 2. Verifying their conjecture, we show that C-n, K-n and K Delta, Delta are the only extremal graphs for this inequality. Confirming a conjecture of Davila and Kenter [5], we show that Z(G) >= 2 delta - 2 for every triangle-free graph G of minimum degree delta >= 2. It is known that Z(G) >= P(G) for every graph G where P(G) is the minimum number of induced paths in G whose vertex sets partition V(G). We study the class of graphs G for which every induced subgraph H of G satisfies Z(H) = P (H). (C) 2016 Elsevier B.V. All rights reserved.