Extremal values and bounds for the zero forcing number

Amos et al. (2015) proved Z(G)≤((Δ−2)n+2)/(Δ−1) for a connected graph G of order n and maximum degree Δ≥2. Verifying their conjecture, we show that Cn, Kn, and KΔ,Δ are the only extremal graphs for this inequality. Confirming a conjecture of Davila and Kenter [5], we show that Z(G)≥2δ−2 for every triangle-free graph G of minimum degree δ≥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).