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Zero forcing in a graph refers to the evolution of vertex states under repeated application of a color change rule. Typically the states are chosen to be blue and white, and a forcing set is an initial set of blue vertices such that all of…

Combinatorics · Mathematics 2025-11-21 Daniela Ferrero , H. Tracy Hall , Leslie Hogben , Mark Hunnell , Ben Small

Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either white or black, and vertex $v$ is a black vertex with only one white neighbor $w$, then…

Combinatorics · Mathematics 2014-10-21 Leslie Hogben , My Huynh , Nicole Kingsley , Sarah Meyer , Shanise Walker , Michael Young

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$ to be colored (black) after some times. "color change rule": a white vertex is changed to a…

Combinatorics · Mathematics 2017-02-23 M. Khosravi , S. Rashidi 2 , A. Sheikhhosseni

Zero forcing is a coloring game played on a graph that was introduced more than ten years ago in several different applications. The goal is to color all the vertices blue by repeated use of a (deterministic) color change rule.…

Combinatorics · Mathematics 2018-12-31 Jesse Geneson , Leslie Hogben

Zero forcing is an iterative graph coloring process studied for its wide array of applications. In this process, the vertices of the graph are initially designated as blue or white, and a zero forcing set is a set of initially blue vertices…

Combinatorics · Mathematics 2026-03-23 Asher Brown , Mark Hunnell , Za'Kiyah Toomer-Sanders , Sarah Weber

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the…

Discrete Mathematics · Computer Science 2017-02-06 Boris Brimkov , Caleb C. Fast , Illya V. Hicks

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time…

Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored under a specific color change rule. The goal is to find a…

Combinatorics · Mathematics 2017-09-27 Franklin H. J. Kenter , Jephian C. -H. Lin

The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of…

Combinatorics · Mathematics 2018-12-18 Deepak Bal , Patrick Bennett , Sean English , Calum MacRury , Paweł Prałat

Zero forcing is a binary coloring game on a graph where a set of filled vertices can force non-filled vertices to become filled following a color change rule. In 2008, the zero forcing number of a graph was shown to be an upper bound on its…

Combinatorics · Mathematics 2025-08-12 Thomas R. Cameron , Jonad Pulaj

Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as…

Discrete Mathematics · Computer Science 2016-04-05 Boris Brimkov , Randy Davila

Zero forcing is a graph coloring process that was defined as a tool for bounding the minimum rank and maximum nullity of a graph. It has also been used for studying control of quantum systems and monitoring electrical power networks. One of…

Combinatorics · Mathematics 2020-05-18 P. A. CrowdMath

The concept of zero forcing involves a dynamic coloring process by which blue vertices cause white vertices to become blue, with the goal of forcing the entire graph blue while choosing as few as possible vertices to be initially blue. Past…

Combinatorics · Mathematics 2024-09-10 Sara Anderton , Kanno Mizozoe , Houston Schuerger , Andrew Schwartz

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices…

Combinatorics · Mathematics 2019-06-27 Yu Chan , Emelie Curl , Jesse Geneson , Leslie Hogben , Kevin Liu , Issac Odegard , Michael S. Ross

Zero forcing is a graph propagation process for which vertices fill-in (or propagate information to) neighbor vertices if all neighbors except for one, are filled. The zero-forcing number is the smallest number of vertices that must be…

Combinatorics · Mathematics 2024-10-24 Heather LeClair , Tim Spilde , Sarah Anderson , Brenda Kroschel

Zero forcing is a deterministic iterative graph coloring process in which vertices are colored either blue or white, and in every round, any blue vertices that have a single white neighbor force these white vertices to become blue. Here we…

Combinatorics · Mathematics 2019-09-17 Sean English , Calum MacRury , Pawel Pralat

Probabilistic zero-forcing is a coloring process on a graph. In this process, an initial set of vertices is colored blue, and the remaining vertices are colored white. At each time step, blue vertices have a non-zero probability of forcing…

Combinatorics · Mathematics 2020-10-26 David Hu , Alec Sun

The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbour colour forcing rule. The zero forcing number provides topological information about linear algebra on graphs,…

Combinatorics · Mathematics 2021-02-10 Alexei Vazquez

Zero forcing is a deterministic iterative graph colouring process in which vertices are coloured either blue or white, and in every round, any blue vertices that have a single white neighbour force these white vertices to become blue. Here…

Combinatorics · Mathematics 2021-03-17 Natalie C. Behague , Trent Marbach , Pawel Pralat

Zero forcing in graphs is a coloring process where a colored vertex can force its unique uncolored neighbor to be colored. A zero forcing set is a set of initially colored vertices capable of eventually coloring all vertices of the graph.…

Combinatorics · Mathematics 2024-05-03 Krishna Menon , Anurag Singh
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