Connected zero forcing sets and connected propagation time of graphs

• Published in: Transactions on combinatorics Vol. 9; no. 2; pp. 77 - 88
• Main Authors: Maryam Khosravi, Saeedeh Rashidi, Alemeh Sheikhhosseini
• Format: Journal Article
• Language: English
• Published: University of Isfahan 01.06.2020
• Subjects: connected zero forcing number; propagation time; zero forcing number
• ISSN: 2251-8657; 2251-8665
• DOI: 10.22108/toc.2020.115286.1617

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.