Fractional zero forcing via three-color forcing games

An r-fold analogue of the positive semidefinite zero forcing process that is carried out on the r-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We also present a three-color interpretation of the skew zero forcing game. The treatment of fractional positive semidefinite forcing number is paralleled to develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number. The threecolor approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.