Computing the zero forcing number for generalized Petersen graphs

Abstract

Let $G$ be a simple undirected graph with each vertex colored
 either white or black, $ u $ be a black vertex of $ G, $ and
 exactly one neighbor $ v $ of $ u $ be white. Then change the
 color of $ v $ to black. When this rule is applied, we say $ u $
 forces $ v, $ and write $ u \rightarrow v $. A $zero\ forcing\ set$ of a graph $ G$ is a
 subset $Z$ of vertices such that if initially the vertices in $ Z $ are colored
 black and remaining vertices
 are colored white, the entire graph $ G $ may be colored black
 by repeatedly applying the
 color-change rule.
 The zero forcing number of $ G$, denoted $Z(G), $ is the minimum size of a
 zero forcing set.\\
 In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by $P(n,k)$). We obtain upper and lower bounds for the zero forcing number for $P(n,k)$. We show that $Z(P(n,2))=6$ for $n\geq 10$, $Z(P(n,3))=8$ for $n\geq 12$ and $Z(P(2k+1,k))=6$ for $k\geq 5$.