Positive semidefinite propagation time

Abstract. Let G be a simple, undirected graph. Positive semidefinite (PSD) zero forcing on G is based on the following 1 color-change rule: Let W1,W2, . . . ,Wk be the sets of vertices of the k connected components in G − B (where B is a set of blue 2 vertices). If w ∈Wi is the only white neighbor of some b ∈ B in the graph G[B∪Wi], then we change w to blue. A minimum positive 3 semidefinite zero forcing set (PSDZFS) is a set of blue vertices that colors the entire graph blue and has minimum cardinality. The 4 PSD propagation time of a PSDZFS B of graph G is the minimum number of iterations that it takes to color the entire graph 5 blue, starting with B blue, such that at each iteration as many vertices are colored blue as allowed by the color-change rule. The 6 minimum and maximum PSD propagation times are taken over all minimum PSD zero forcing sets of the graph. It is conjectured 7 that every propagation time between the minimum and maximum propagation time is attainable by some minimum PSDZFS (this 8 is not the case for the standard color-change rule). Tools are developed that aid in the computation of PSD propagation time. 9 Several graph families and graphs with extreme PSD propagation times (|G| − 2, |G| − 1, 1, 0) are analyzed. 10