On the Zero Forcing Number of Trees

Abstract

Let G be a graph such that the color of its vertices is white or black. A dynamic vertex coloring for G is defined as follows. One starts with a certain set of black vertices. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a zero forcing set if by iterating this process, all of the vertices of G become black. The zero forcing number of G (denoted by Z(G)) is the minimum cardinality of a zero forcing set in G. In this paper, we study the zero forcing number of trees. Let T be a tree with at least two vertices. We show that $\Delta (T)-1\le Z(T)\le r(T)-1$, where $\Delta (T)$ and r(T) are the maximum degree and the number of pendant vertices of T, respectively. As a consequence, we obtain that $Z(L(T))\ge Z(T)$, where L(T) is the line graph of T. We characterize all trees T such that $Z(T)=\Delta (T)-1$. Finally, we study trees T with $Z(T)= r(T)-1$.