# Zero forcing number of graphs

## TL;DR

This paper investigates the forcing number, a measure of the minimum initial infected vertices needed to infect an entire graph through a specific process, across various graph classes including large girth, $H$-free, random, and pseudorandom graphs.

## Contribution

It provides new insights and bounds on the forcing number for multiple classes of graphs, expanding understanding of infection processes in graph theory.

## Key findings

- Determines bounds for large girth graphs.
- Analyzes forcing number in $H$-free graphs.
- Studies behavior in random and pseudorandom graphs.

## Abstract

A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of $G$ is the minimum cardinality of a forcing set in $G$. In the present paper, we study the forcing number of various classes of graphs, including graphs of large girth, $H$-free graphs for a fixed bipartite graph $H$, random and pseudorandom graphs.