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Throttling for zero forcing and variants. (English) Zbl 1437.05070
Author’s abstract: Zero forcing is a process on a graph in which the goal is to force all vertices to become blue by applying a color change rule. Throttling minimizes the sum of the number of vertices that are initially blue and the number of time steps needed to color every vertex. This paper provides a new general definition of throttling for variants of zero forcing and studies throttling for the minor monotone floor of zero forcing. The technique of using a zero forcing process to extend a given graph is introduced. For standard zero forcing and its floor, these extensions are used to characterize graphs with throttling number at most $$t$$ as certain minors of Cartesian products of complete graphs and paths. Finally, these characterizations are applied to determine graphs with extreme throttling numbers.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C76 Graph operations (line graphs, products, etc.) 05C35 Extremal problems in graph theory 05C85 Graph algorithms (graph-theoretic aspects)
##### Keywords:
extreme throttling numbers
Full Text:
##### References:
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