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 → v. A zero f orcing 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 ≥ 10, Z(P (n, 3)) = 8 for n ≥ 12 and Z(P (2k + 1, k)) = 6 for k ≥ 5.