# Connected zero forcing sets and connected propagation time of graphs

## TL;DR

This paper explores the concept of connected zero forcing sets and propagation times in graphs, characterizing graphs with extreme propagation times and analyzing special graph classes and products.

## Contribution

It introduces the connected zero forcing set and propagation time, providing characterizations for graphs with extreme propagation times and analyzing specific graph classes.

## Key findings

- Graphs with propagation time |G|-1 and |G|-2 are characterized.
- Connected zero forcing sets are studied for special graphs and graph products.
- New concepts of connected zero forcing and propagation time are introduced.

## Abstract

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 black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs with extreme minimum connected propagation times and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.