On Zero Forcing Number of Graphs and Their Complements

Abstract

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G) \setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank -- Special Graphs Work Group". It's known that $Z(G)\geq δ(G)$, where $δ(G)$ is the minimum degree of $G$. We show that $Z(G)\leq n-3$ if a connected graph $G$ of order $n$ has a connected complement graph $\overline{G}$. Further, we characterize a tree or a unicyclic graph $G$ which satisfies either $Z(G)+Z(\overline{G})=δ(G)+δ(\overline{G})$ or $Z(G)+Z(\overline{G})=2(n-3)$.