The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become...

Full description

Saved in:
Bibliographic Details
Published inDiscrete Applied Mathematics Vol. 258; pp. 35 - 48
Main Authors Boyer, Kirk, Brimkov, Boris, English, Sean, Ferrero, Daniela, Keller, Ariel, Kirsch, Rachel, Phillips, Michael, Reinhart, Carolyn
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 15.04.2019
Elsevier BV
Subjects
Apexes
Graph coloring
Graph theory
Polynomials
Recognizability
Threshold graph
Unimodality
Zero forcing polynomial
Zero forcing set
Unimodality
Zero forcing polynomial
Threshold graph
Recognizability
Zero forcing set
Online AccessGet full text

Cover

More Information
Summary:Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph G of order n as the polynomial Z(G;x)=∑i=1nz(G;i)xi, where z(G;i) is the number of zero forcing sets of G of size i. We characterize the extremal coefficients of Z(G;x), derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of Z(G;x), including multiplicativity, unimodality, and uniqueness.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2018.11.033