Propagation time for zero forcing on a graph

TL;DR

This paper studies the propagation time in zero forcing on graphs, characterizing extremal cases, and exploring relationships between propagation time, graph diameter, and structural properties.

Contribution

It characterizes graphs with extreme minimum propagation times and analyzes the relationship between propagation time and graph diameter.

Findings

Connected graphs have multiple minimum zero forcing sets with minimal propagation time.

Graphs with minimum propagation times of |G|-1, |G|-2, and 0 are characterized.

Diameter bounds the maximum propagation time for trees, but not in general.

Abstract

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 change the color of $w$ to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color change rule. The propagation time of a zero forcing set $B$ of graph $G$ is the minimum number of steps that it takes to force all the vertices of $G$ black, starting with the vertices in $B$ black and performing independent forces simultaneously. The minimum and maximum propagation times of a graph are taken over all minimum zero forcing sets of the graph. It is shown that a connected graph of order at least two has more than one minimum zero forcing set realizing minimum propagation time. Graphs $G$ having extreme minimum propagation times $|G| - 1$, $|G| - 2$, and $0$ are characterized, and results regarding graphs having minimum propagation time $1$ are established. It is shown that the diameter is an upper bound for maximum propagation time for a tree, but in general propagation time and diameter of a graph are not comparable.