2014/10/15 by Leslie Hogben, My Huynh, Nicole Kingsley +3 · 1 citation
Mathematics · #math.CO
paper · pdf · doi:10.1016/j.dam.2012.04.003
Poster Presentation Presented at USTARS 2012
arxiv created 2014/10/15 · arxiv updated 2014/10/21
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.