Iteration Index of a Zero Forcing Set in a Graph

TL;DR

This paper introduces the iteration index of a zero forcing set in a graph, a new invariant measuring the number of steps needed to turn all vertices black, with initial properties and results for specific graphs.

Contribution

It defines the iteration index of a zero forcing set as a new graph invariant and explores its basic properties and initial results for certain graph classes.

Findings

Defined the iteration index as a new graph invariant.

Established basic properties of the iteration index.

Presented preliminary results for specific graphs.

Abstract

Let each vertex of a graph G = (V(G), E(G)) be given one of two colors, say, "black" and "white". Let Z denote the (initial) set of black vertices of G. The color-change rule converts the color of a vertex from white to black if the white vertex is the only white neighbor of a black vertex. The set Z is said to be a zero forcing set of G if all vertices of G will be turned black after finitely many applications of the color-change rule. The zero forcing number of G is the minimum of |Z| over all zero forcing sets Z \subseteq V (G). Zero forcing parameters have been studied and applied to the minimum rank problem for graphs in numerous articles. We define the iteration index of a zero forcing set of a graph G to be the number of (global) applications of the color-change rule required to turn all vertices of G black; this leads to a new graph invariant, the iteration index of G - it is the minimum of iteration indices of all minimum zero forcing sets of G. We present some basic properties of the iteration index and discuss some preliminary results on certain graphs.