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Propagation time for zero forcing on a graph. (English) Zbl 1246.05056
Summary: 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
zero forcing number; propagation time
Full Text:
##### References:
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