Positive semidefinite zero forcing

The positive semidefinite zero forcing number _Z+(G)${\mathrm{Z}}_{+}(G)$ of a graph G was introduced in Barioli et al. (2010) 4]. We establish a variety of properties of _Z+(G)${\mathrm{Z}}_{+}(G)$: Any vertex of G   can be in a minimum positive semidefinite zero forcing set (this is not true for standard zero forcing). The graph parameters tw(G)$\mathrm{tw}(G)$ (tree-width), _Z+(G)${\mathrm{Z}}_{+}(G)$, and Z(G)$\mathrm{Z}(G)$ (standard zero forcing number) all satisfy the Graph Complement Conjecture (see Barioli et al. (2012) 3]). Graphs having extreme values of the positive semidefinite zero forcing number are characterized. The effect of various graph operations on positive semidefinite zero forcing number and connections with other graph parameters are studied.