Zero forcing parameters and minimum rank problems

Abstract

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G| − OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hemitian positive semidefinite minimum rank is presented.

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

“…Zero forcing for positive semidefinite matrices was defined in [3], using the following color change rule.…”

Induced trees, minimum semidefinite rank, and zero forcing

“…Defined by the AIM Minimum Rank-Special Graphs Work Group [AIM 2008], zero forcing was also the result of looking for approaches to solving a minimum rank problem, but has since been shown to be of interest in quantum physics [Burgarth et al 2011]. It turns out that the OS number and the positive semidefinite zero forcing number are two sides of the same coin, as for any graph they sum to the number of vertices [Barioli et al 2010]. Moreover, the complement of an OS set is a zero forcing set and vice versa.…”

Zero forcing number, path cover number, and maximum nullity of cacti

“…The zero forcing number of a graph was introduced in [AIM 2008] and the related terminology was developed in [Barioli et al 2009], [Barioli et al 2010], and [Hogben 2010]. Referring to it as the graph infection number, physicists have used this parameter in studying quantum systems control [Burgarth and Giovannetti 2007;Burgarth and Maruyama 2009;Severini 2008].…”

“…When a chain set contains a chain consisting of a single vertex, we say that the chain set contains the vertex as a singleton. For a zero forcing set Z , a reversal of Z is the set of vertices which are last in the forcing chains in the chain set of some chronological list of forces [Barioli et al 2010]. Theorem 1.3 [Barioli et al 2010].…”

“…For a zero forcing set Z , a reversal of Z is the set of vertices which are last in the forcing chains in the chain set of some chronological list of forces [Barioli et al 2010]. Theorem 1.3 [Barioli et al 2010]. If Z is a zero forcing set of G then so is any reversal of Z .…”