Minimum rank and failed zero forcing number of graphs

TL;DR

This paper investigates the failed zero forcing number in graphs under standard and positive semidefinite rules, relating it to the minimum rank and characterizing specific graph families.

Contribution

It introduces the concept of failed zero forcing number, computes it for various graph families, and characterizes graphs where this number equals the minimum rank.

Findings

Failed zero forcing number computed for several graph families.

Characterization of graphs where failed zero forcing number equals minimum rank.

Analysis under both standard and positive semidefinite color-change rules.

Abstract

Let $G$ be a simple, finite, and undirected graph with vertices each given an initial coloring of either blue or white. Zero forcing on graph $G$ is an iterative process of forcing its white vertices to become blue after a finite application of a specified color-change rule. We say that an initial set $S$ of blue vertices of $G$ is a zero forcing set for $G$ under the specified color-change rule if a finite number of iterations of zero forcing results to an updated coloring where all vertices of $G$ are blue. Otherwise, we say that $S$ is a failed zero forcing set for $G$ under the specified color-change rule. It is not difficult to see that any subset of a failed zero forcing set is also failed. Hence, our interest lies on the maximum possible cardinality of a failed zero forcing set, which we refer to as the failed zero forcing number of $G$. In this paper, we consider two color-change rules $-$ standard and positive semidefinite. We compute for the failed zero forcing numbers of several graph families. Furthermore, under each graph family, we characterize the graphs $G$ for which the failed zero forcing number is equal to the minimum rank of $G$.